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 2019-11-06, 13:46 #1 enzocreti   Mar 2018 72·11 Posts binary form of the exponents 69660, 92020, 541456 pg(69660), pg(92020) and pg(541456) are probable primes with 69660, 92020 and 541456 multiple of 86 69660 in binary is 10001000000011100 92020 in binary is 10110011101110100 541456 in binary is 10000100001100010000 you can see that the number of the 1's is always a multiple of 5 a chance?
 2019-12-18, 08:26 #2 enzocreti   Mar 2018 53910 Posts 69660, 92020, 541456 69660, 92020 and 541456 are 6 mod 13 (and 10^m mod 41) 69660, 92020 and 541456 are multiple of 43 is there a reason why (69660-6)/26 is congruent to 13 mod 43 (92020-6)/26 is congruent to 13 mod 43 (541456-6)/26 is congruent to 13 mod 43? 215, 69660, 92020, 541456 are multiple of 43 let be log the log base 10 int(x) let be the integer part of x so for example int(5.43)=5 A=10^2*log(215)-215/41 int(A)=227=B 215 (which is odd) is congruent to B+1 mod 13 69660 which is even is congruent to B mod 13 92020 is congruent to B mod 13 and 541456 is also congruent to B mod 13 215 (odd) is congruent to -1215 mod 13 69660 (even) is congruent to 1215 mod 13 92020 (even) is congruent to 1215 mod 13 541456 (even) is congruent to 1215 mod 13 215, 69660, 92020, 541456 can be written as 13x+1763y+769 (541456-769-13*93)/1763=306 (69660-769-13*824)/1763=33 (92020-769-13*781)/1763=46 as you can see 306,33 and 46 are all 7 mod 13 769+13*824 769+13*781 769+13*93 are multiples of 43 (69660-(10^3+215))/13=5265 which is 19 mod 43 (92020-(10^3+215))/13=6985 which is 19 mod 43 (541456-(10^3+215))/13=41557 which is 19 mod 43 10^3+215 is 6 mod 13 now 69660/13=5358,4615384... 92020/13=7078,4615384... 541456/13=41650,4615384... the repeating term 4615384 is the same...so that numbers must have some form 13s+k? 215 (odd) is congruent to 307*2^2-10^3 or equivalently to - (19*2^6-1) mod 13 69660 (even) is congruent to 307*2^2-1001 or equivalenly to (19*2^6-1) mod 13 92020 (even) the same 541456 (even) the same (19*2^6*(541456-92020)/(13*43)+1)/13+10=75215 is a multiple of 307=(215*10-1)/7=(5414560-1)/17637 pg(51456) is another probable prime with 51456 congruent to 10^n mod 41 75215=(51456*19+1)/13+10=(19*2^6*(541456-92020)/(13*43)+1)/13+10 215 is congruent to -1215=5*3^5 mod 13 69660 is congruent to 1215 mod 13 and so also 92020 and 541456 1215=(51456/2-2*13*10^2-43)/19 ...so summing up... 215 (odd) is congruent to 3*19*2^2 mod((41*43-307)/(7*2^4)=13) where 41*43-307 is congruent to 10^3+3*19*2^2 mod (3*19*2^2=228) 69660 (even) is congruent to (3*19*2^2-1) mod 13 ... the same for 92020 and 541456 another way is 215 (odd) is congruent to -15*81 mod ((41*43-307)/(307-15*13)) 69660 (even) is congruent to 15*81 mod ((41*43-307)/(307-15*13)) the same for 92020 and 541456 215 is congruent to (41*43-307-10^3)/2 mod ((41*43-307)/(7*2^4)) 69660 is congruent to (41*43-307-10^3)/2-1 mod ((41*43-307)/(7*2^4)) 92020 is congruent to (41*43-307-10^3)/2-1 mod ((41*43-307)/(7*2^4)) 541456 is congruent to (41*43-307-10^3)/2-1 mod ((41*43-307)/(7*2^4)) 215 is congruent to 3*19*2^2 mod ((41*43-307)/(2*(19*3-1))) 69660 is congruent to 3*19*2^2-1 mod ((41*43-307)/(2*(19*3-1))) and so 92020 and 541456 215 is congruent to 3*19*2^2 mod ((3^6-1)/(3*19-1)) 69660 is congruent to 3*19*2^2-1 mod((3^6-1)/(3*19-1)) and so 92020 and 541456 51456 (pg(51456) is probable prime and 51456 is 10^n mod 41) is congruent to 19*3*2^4 mod 13 Pg(2131) is probable prime 2131 is prime 227=2131-307*4-26^2 So 215 is congruent also to 2131-307*4-26^2 mod 13 And 69660 is congruent to 2131-307*4-26^2+1 mod 13 and so also 92020 and 541456 69660 is congruent to 1763-307*5-1 mod ((1763-307)/112)=13) And so 92020 and 541456 215 which is odd is congruent to - 1763+307*5+1 mod 13 215+1763-307*5-1 is divisible by 17 and by 13 And also 541456-1763+307*5+1 is divisible by 13 and 17 215 and 541456 have the same residue 10 mod 41 ((541456-1763+307*5+1)/(13*17)+1) *200+51456=541456 69660 92020 541456 are congruent to 7*2^6 mod 26 7*2^6-(1763-307*5-1)=221=13*17 215+7*2^6 is a multiple of 221 541456-7*2^6 is a multiple of 221 Last fiddled with by enzocreti on 2020-01-26 at 17:58
 2020-01-24, 15:31 #3 enzocreti   Mar 2018 21B16 Posts 215 , 51456, 69660, 92020, 541456 51456, 69660, 92020, 541456 are even and congruent to 10^n mod 41 pg(51456), pg(69660), pg(92020) and pg(541456) are prp 51456, 69660, 92020, 541456 are all congruent to 7*2^q-1 mod 13 with q a nonnegative integer 215 is odd and pg(215) is prp 215 is congruent to 7*2^q mod 13
 2020-03-30, 11:52 #4 enzocreti   Mar 2018 72×11 Posts 69660 and 92020 69660 and 92020 are multiple of 215 and congruent to 344 mod 559 92020=lcm(215,344,559)+69660 | denotes concatenation in base 10 2^69660-1 | 2^69559-1 is prime 2^92020-1 | 2^92019-1 is prime!!! Last fiddled with by enzocreti on 2020-03-30 at 11:53
 2020-03-30, 12:19 #5 LaurV Romulan Interpreter     "name field" Jun 2011 Thailand 240468 Posts please show us a proof that they are prime
 2020-03-30, 12:50 #6 enzocreti   Mar 2018 72×11 Posts ... Well... actually they are only probable primes... maybe in future they will be proven primes Last fiddled with by enzocreti on 2020-03-30 at 12:52
2020-03-30, 17:39   #7
enzocreti

Mar 2018

53910 Posts

Quote:
 Originally Posted by LaurV please show us a proof that they are prime

http://factordb.com/index.php?id=1100000001110801143

http://factordb.com/index.php?query=...%2B2%5E92019-1

Last fiddled with by enzocreti on 2020-03-30 at 17:55

 2020-03-30, 23:06 #8 enzocreti   Mar 2018 10000110112 Posts ... I note also... 69660 I note also that (lcm(215,344,559))^2=4999*10^5+69660-60 I notice that lcm(215,344,559)=22360 22360/(18*18)=69.01234567... curious I notice also that 541456 (multiple of 43),is - 215 mod (18*18-1) and 69660 (multiple of 43) is 215 mod (18*18-1) I note that the polynomial X^2-X*429^2+7967780460=0 has the solution x=69660 If you see the discriminant of such polynomial you can see interesting things about pg primes with exponent multiple of 43 I note that 429^2 is congruent to 1 mod 215 and to 1 mod 344. I note that 92020*2+1=429^2 The discriminant of the polynomial is 429^4-4*7967780460 which is a perfect square and lcm(215,344,559) divides 429^4-4*7967780460-1 Pg(331259) is prime and pg(92020) is prime. 92020+(92020/215-1)*559+546=331259 Again magic numbers 559 and 546 strike! So 331259 is a number of the form 215*(13s-1)+(13*b-2)*559+546 For some s, b positive integers So there are pg primes pg(75894) and pg(56238) with 75894 and 56238 multiple of 546 and pg(331259) with 331259 of the form 215m+559g+546 for some positive m and g. Pg(69660) is prime. 69660=(3067*8-546)*3-11# where # is the primorial and 3067 is a prime of the form 787+456s notice that lcm(215,344,559)=22360 22360/(18*18)=69.01234567... curious I notice also that 541456 (multiple of 43),is - 215 mod (18*18-1) and 69660 (multiple of 43) is 215 mod (18*18-1) So we have pg(215) is prime pg(69660) is prime pg(92020) is prime With 215 69660 and 92020 multiple of 43 69660=215+(18*18-1)*215 92020=69660+18*18*69+4 -215 is congruent to 108 mod (18*18-1=323) 541456 is congruent to 108 mod (18*18-1) 92020 is congruent to (17*17-1) mod (18*18-1) 69660 is congruent to 215 mod (18*18-1) 108=6^3-18^2 17*17-1+36-216=17*17-1+6^2-6^3=108 215=6^3-1 To make it easier 215, 69660, 541456 are congruent to plus or minus 215 mod 323 92020 is congruent to (17*17-1) mod (18*18-1) curious that 289/215 is about 1.(344)... and 541456 92020 69660 215 are congruent to plus or minus 344 mod 559 215, 69660, 541456 are congruent to plus or minus 6^3-1 mod 323 92020 is congruent to (12/9)*6^3 mod 323 92020*9/12 is congruent to 6^3 mod 323 92020 is congruent to (17^2-1) mod 323 and to - (6^2-1) mod 323 92020 is a number of the form 8686+13889s 13889=(6^3+1)*64 215 69660 92020 541456 are + or - 344 mod 559 lcm(215,344,559)-86*(10^2+1)+6^3-1=(6^3+1)*2^6+1 92020=69660+lcm(215,344,559) so you can substitute 92020=69660+86*(10^2+1)-6^3+1+(6^3+1)*2^6+1 86*(10^2+1) mod 323 is 17^2-1 215, 69660, 541456 are multiple of 43 and congruent to 10 and 1 mod 41 They are congruent to plus or minus 215 mod 323 92020 is congruent to 2^4 (not a power of 10) mod 41 92020 is congruent to (2^4+1)^2-1 mod 323 288 is 17^2-1 288 in base 16 is 120 120=11^2-1 also 323=18^2-1 in base 16 is 143=12^2-1 344*((1444456-1763*2^9) /344-1)=541456 1444456=lcm(13,323,344) 541456=lcm(13,323,344)-344*(41*2^6+1) 215 69660 92020 541456 are congruent to plus or minus (3^a*2^b) mod 323 215 is congruent to - 108 mod 323 541456 is congruent to 108 mod 323 69660 is congruent to - 108 mod 323 92020 is congruent to 288 mod 323 108 and 288 are numbers of the form 3^a*2^b So exponents multiple of 43 are congruent to plus or minus 344 mod 559 and to plus or minus a 3-smooth number mod 323 108 and 288 are both divisible by 36 Last fiddled with by enzocreti on 2020-08-18 at 21:00 Reason: notice that lcm(215,344,559)=22360 22360/(18*18)=69.01234567... curious I notice also that 541456 (multiple of 43),is - 21
2020-08-19, 05:48   #9
enzocreti

Mar 2018

72·11 Posts

Q77I
Quote:
 Originally Posted by enzocreti 69660 I note also that (lcm(215,344,559))^2=4999*10^5+69660-60 I notice that lcm(215,344,559)=22360 22360/(18*18)=69.01234567... curious I notice also that 541456 (multiple of 43),is - 215 mod (18*18-1) and 69660 (multiple of 43) is 215 mod (18*18-1) I note that the polynomial X^2-X*429^2+7967780460=0 has the solution x=69660 If you see the discriminant of such polynomial you can see interesting things about pg primes with exponent multiple of 43 I note that 429^2 is congruent to 1 mod 215 and to 1 mod 344. I note that 92020*2+1=429^2 The discriminant of the polynomial is 429^4-4*7967780460 which is a perfect square and lcm(215,344,559) divides 429^4-4*7967780460-1 Pg(331259) is prime and pg(92020) is prime. 92020+(92020/215-1)*559+546=331259 Again magic numbers 559 and 546 strike! So 331259 is a number of the form 215*(13s-1)+(13*b-2)*559+546 For some s, b positive integers So there are pg primes pg(75894) and pg(56238) with 75894 and 56238 multiple of 546 and pg(331259) with 331259 of the form 215m+559g+546 for some positive m and g. Pg(69660) is prime. 69660=(3067*8-546)*3-11# where # is the primorial and 3067 is a prime of the form 787+456s notice that lcm(215,344,559)=22360 22360/(18*18)=69.01234567... curious I notice also that 541456 (multiple of 43),is - 215 mod (18*18-1) and 69660 (multiple of 43) is 215 mod (18*18-1) So we have pg(215) is prime pg(69660) is prime pg(92020) is prime With 215 69660 and 92020 multiple of 43 69660=215+(18*18-1)*215 92020=69660+18*18*69+4 -215 is congruent to 108 mod (18*18-1=323) 541456 is congruent to 108 mod (18*18-1) 92020 is congruent to (17*17-1) mod (18*18-1) 69660 is congruent to 215 mod (18*18-1) 108=6^3-18^2 17*17-1+36-216=17*17-1+6^2-6^3=108 215=6^3-1 To make it easier 215, 69660, 541456 are congruent to plus or minus 215 mod 323 92020 is congruent to (17*17-1) mod (18*18-1) curious that 289/215 is about 1.(344)... and 541456 92020 69660 215 are congruent to plus or minus 344 mod 559 215, 69660, 541456 are congruent to plus or minus 6^3-1 mod 323 92020 is congruent to (12/9)*6^3 mod 323 92020*9/12 is congruent to 6^3 mod 323 92020 is congruent to (17^2-1) mod 323 and to - (6^2-1) mod 323 92020 is a number of the form 8686+13889s 13889=(6^3+1)*64 215 69660 92020 541456 are + or - 344 mod 559 lcm(215,344,559)-86*(10^2+1)+6^3-1=(6^3+1)*2^6+1 92020=69660+lcm(215,344,559) so you can substitute 92020=69660+86*(10^2+1)-6^3+1+(6^3+1)*2^6+1 86*(10^2+1) mod 323 is 17^2-1 215, 69660, 541456 are multiple of 43 and congruent to 10 and 1 mod 41 They are congruent to plus or minus 215 mod 323 92020 is congruent to 2^4 (not a power of 10) mod 41 92020 is congruent to (2^4+1)^2-1 mod 323 288 is 17^2-1 288 in base 16 is 120 120=11^2-1 also 323=18^2-1 in base 16 is 143=12^2-1 344*((1444456-1763*2^9) /344-1)=541456 1444456=lcm(13,323,344) 541456=lcm(13,323,344)-344*(41*2^6+1) 215 69660 92020 541456 are congruent to plus or minus (3^a*2^b) mod 323 215 is congruent to - 108 mod 323 541456 is congruent to 108 mod 323 69660 is congruent to - 108 mod 323 92020 is congruent to 288 mod 323 108 and 288 are numbers of the form 3^a*2^b So exponents multiple of 43 are congruent to plus or minus 344 mod 559 and to plus or minus a 3-smooth number mod 323 108 and 288 are both divisible by 36

215 69660 92020 541456 are congruent to plus or minus (6^k-1) mod 323 for k=3,2

69660=(2^5*3^7)-(2^3*3^4) so it is the difference of two 3 smooth numbers (2^a*3^b)-(2^(a-3)*3^(b-3))

69660 is multiple of 3 and congruent to 0 mod (6^2-1)

215, 92020, 541456 are not multiple of 3 and multiple of 43 and are congruent to plus or minus 2^k mod 36 for some k

1763*323-(6^3+1)*(2^7+1)=541456 or

(42^2-1)*(18^2-1)-(6^3+1)*(2^7+1)=541456

I also note that

69660=(2^7+1)*(6^3+1)+(2^7+1)*(18^2-1)

And by the way 215=(42^2-1)*4^2-(2^7+1)*(6^3+1)

27993=(217*129)=(2^7+1)*(6^3+1)

I notice that - 541456 mod 27993=9202*2

92020=10*9202

And 9202*2*10+1=429^2

27993=3/5*(6^6-1)

27993 in base 6 is 333333

27993 has also the representation:

(42^2-1)*4^2-(6^3-1)=27993=2*43*(18^2-1)+(6^3-1)

x/5+(42^2-1)*(18^2-1)-3/5*(6^6-1)=20*3/5*(6^6-1)

The solution of this equation x=92020

this identity:

(42^2-1)*(18^2-1)=(10^3+18^2)*430+43*3

Maybe it is not a chance that pg(10^3+18^2-1=1323) is prime

pg(1323), pg(215), pg(69660), pg(92020), pg(541456) are primes

1323, 215, 69660, 92020, 541456 are congruent to plus or minus (2^a*3^b-1) mod 323 where 2^a*3^b is a 3 smooth number with 2^a*3^b<323

or 1323, 215, 69660, 92020, 541456 are congruent to plus or minus (p-1) mod 323 where p is a perfect power

92020 has the factorization 2*(6^3-1)*(6^3-2)

lcm(215,344,559)=(6^3-1)*(2*(6^3-2)-18^2)=(2*(6^3-2)-18^2)*(2*(6^3-2)-18^2+111)

69660=92020-lcm(215,344,559)

215=2*(6^3-2)+2*(6^3+1)-2*18^2+1

92020=(2*(6^3-2)+2*(6^3+1)-2*18^2+1)*(2*(6^3-2)+2*(6^3+1)-2*18^2)*2

so 215=2*(6^3-2)+2*(6^3+1)-2*18^2+1

69660==(2*(6^3-2)+2*(6^3+1)-2*18^2+1)*(2*(6^3-2)+2*(6^3+1)-2*18^2)*2-((6^3-1)*(2*(6^3-2)-18^2))

92020=(2*(6^3-2)+2*(6^3+1)-2*18^2+1)*(2*(6^3-2)+2*(6^3+1)-2*18^2)*2

541456=(42^2-1)*(18^2-1)-(7^3-6^3+2)*(6^3+1)

69660=111111-5*11111+8*(42^2-1)

344=7^3+1=2*(111111-5*11111)/(18^2-1)=A

so lcm(215,344,559)=lcm(215, A, A+215)

541456=5*111112-(42^2-1)*8

(12^2-1) divides (111111-5*11111-6^3+1)

559=(111111-5*11111-6^3+1)/(12^2-1)+(111111-5*11111)/(18^2-1)

Quite clear that 215, 69660, 92020,541456 multiple of 43

are congruent to plus or minus (18^2-6^k) mod (18^2-1) where k is 2 or 3 and 3 indeed is the maximum exponent such that 18^2-6^k i 1s positive

(541456-18^2+6^3)/323=41^2-5=1676
(92020-18^2+6^2)/323=17^2-5=284
(69660+18^2-6^3)/323=6^3
(215+18^2-6^3)/323=1

There is clearly a pattern!!!

I notice also that (1+5)=6 is a semiprime
(6^3+5)=13*17 is a semiprime
(1676+5)=41*41 is a semiprime
(284+5)=17*17 is a semiprime

(6,221,1681,289) are either squares of primes or product of two consecutive primes

so when pg(k) is prime and k is a multiple of 43, then k can be expressed in this way:

(p*q-5)*(18^2-1) plus or minus (18^2-6^k) with 6^k<18^2

p and q are primes

when pg(k) is prime and k is a multiple of 546, then

k is congruent to 78 mod (18^2-6^3)

as in the cases k=56238 and k=75894

Curio of the curios:

pg(331259) is probable prime and 331259 is prime

magic:

71*6^6-331*(10^4-3*331)=331259

71*6^6 and 331259 have in common the first five digits 33125=182^2+1

If you consider 71*6^n for n>3

For n even

71*6^n is congruent to (11^2-1) mod (13^2-1)

and for n odd

Is congruent to (7^2-1) mod (13^2-1)

For n=4

71*6^4=92016

for n=6

71*6^6=3312576

As you can see

Pg(92020) is prime and pg(331259) is prime

92020 has the last two digits 20 different from 92016

The same for 3312576 and 331259

Moreover 331259 mod (71*6^3) =9203

Both 331259 and 92020 are 5 mod 239

Is there something connected with the fact

ord (71*6^k)=4

I mean the smallest value k for which 71*6^k is congruent to 1 mod 239 is k=4???

71 and 6 are both quadratic residues mod 239

92020 and 331259 are congruent to 71*3^3 mod (239*13)

I wonder if this concept could be generalized

pg(51456), pg(92020), pg(331259) are probable primes

51456 is congruent to 71 mod (239*(6^3-1))
92020 is congruent to 71*3^3 mod (239*13)
331259 is congruent to 71*3^3 mod (239*13)

92020 and 331259 are congruent to 6 mod 13

51456 and 331259 are congruent to 2^3 mod 109

71 and 71*3^3 are both congruent to 6 mod 13

so

51456, 92020, 331259 are congruent to 13*(5+71*k)+6 either mod (239*215) or mod (239*13)

are these primes infinite?

the odd thing is that

51456, 92020, 331259 are either congruent to 10^m mod 41 or prime (331259 infact is prime)

So I think it is no chance that 51456, 92020, 331259 are either congruent to 2^j mod 71 or to 13*2^i mod 71

No chance at all!

There is a file Rouge!
Pg(541456) is probable prime as pg(51456)

And 541456 mod (239*215)=9202*3, that is 3*( 92020/10 )and pg(92020) is prime

541456 is congruent to (3/20)*(429^2-1) mod (239*215)

541456 is congruent to 14*71*3^3+3*2^8 (mod (239*215))

Maybe there is some connection to the fact that

215, 69660, 92020 and 541456 are plus or minus 344 mod 559

lcm(344,559)=4472=71*(2^6-1)-1

I suspect that something in field F(239) is in action!

331259^(-1) mod (215*239)=49999=(10^5-2)/2

((71*(2^6-1))-1)/2=lcm(344,559)=9202-6966

Multiplying both sides by 10 you have

92020=69660+lcm(215,344,559)

I would suggest to study these exponents in field F(51385=239*215)

(239*10*215+3*(429^2-1)/10-541456) /3=9202

So

10*
(239*10*215+3*(429^2-1)/10-541456) /3=92020

In this equation we have 215, 92020, 541456 multiple of 43 and not of 3

The other multiple of 43 is 69660 which is multiple of 3

And 92020=69660+lcm(344,215,559)

239*10*215+3*(429^2-1)/10 is a multiple of 559

239*10*215+3*(429^2-1)/10=(42^2-1)*(18^2-2)+43*2^5

This identity

(239*10*215+(3/10)*(429^2-1)-541456+92020)/559=92020/(215*2)

One can play around with this expression containing 215 and 559

And substitute for example 92020 with (429^2-1)/2

Pg(331259) is prime and also 331259 is prime

The inverse modulo (215*239) of 331259 is the prime 5*10^4-1=49999

(331259*49999-1)/(215*239)=7*(18^2-1)*(12^2-1)-10^3+1

So 331259=((((18^2-1)*(10^3+1)-10^3+1)*239*215)+1)/(5*10^4-1)

Using Wolphram Alpha i considered this equation:

((323*(10^x+1)-10^x+1)*239*215+1)/(5*10^(x+1)-1)=y

wolphram say that the integer solution is x=3, y=331259

wolphram gives an alternate form:

84898302/(5*(2^(x+1)*5^(x+2)-1))+1654597/5=y

the number 1654597=69660+(30^2-1)*(42^2-1)

so 69660 and 331259 (both 6 mod 13 and pg(69660) pg(331259) primes) are linked by this equation:

84898302/(5*(2^(3+1)*5^(3+2)-1))+(69660+(30^2-1)*(42^2-1))/5=331259

541456 in field F(239*215) and in field F(239*323)

541456 mod (239*215)=9202*3
541456 mod (239*323)=359*3

Pg(359) is prime, pg(9202*10) is prime

9202-359 is a multiple of 239

Pg(92020+239239=331259) is prime

I note also

331259=71*6^6-331*(10^4-3*331)=6^2*9202-13=92020+239239

239239 is congruent to -13 mod 107 and mod 43

22360=(10*(239239+13))/107

92020=69660+22360

92020=331259-239239

92020 is a multiple of 107

lcm(215,344,559)*10=22360

multiple of 86, that is 69660, 92020 and 541456 are congruent to a square mod 428

so the muliple of 86=k for which pg(k) is prime are numbers for which there is a solution to this modular equation:

y is congruent to 36*x^2 mod 428

infact

92020 is congruent to 36*0^2 mod 428
69660 is congruent to 36*3^2 mod 428
541456 is congruent to 36*1^2 mod 428

There are pg(k) primes with k multiple of 215 and k multiple of 546

probably that numbers 215 and 546 are not random at all

look at this equality:

71*6^6-182^2-3^2-331*(10^4-3*331)-10=546^2

546-215=331=546-(6^3-1)

by the way 182^2+3^2+546^2 is prime

546^2 is congruent to 6^3 mod 331

541456-((3*239) ^2-239+(429^2-1)/20)=(429^2-1)/10

541456=(18/5)*(429^2-1)/2+13*((429^2-11)/10-2235)

where 2235=lcm(215,344,559)-1

541456*10=3*331259+239*(136^2+1)

92020 and 331259 are both 5 mod 239

71*6^k is congruent to 1 mod 239 for k=4

But 71*6^4 is 920...
And 71*6^6 is 33125...

92020 mod 71 is 4 and mod 331 is 3

71*6^4 is congruent to 1 mod (385*239)

541456 is congruent to - 239 mod 385

I notice that 92020 and 331259 are congruent to 5 mod 239 but also to - 72 mod (1001)

I notice that 92020 is congruent to 146 mod 71

541456 and 331259 are congruent to 146 mod 703

I notice that 215, 69660, 541456 are plus or minus 215 mod 323
92020 which is congruent to 16 mod 41, is congruent to 288 mod 323 and mod 71
288 and 92020 are both 4 mod 71

so 92020 is congruent to 4 mod 71
and is congruent to 4+284 (mod 323)

where 284 is the residue mod 323 of 71*6^4

215, 69660, 541456 are congruent to plus or minus 215 mod 323

where 215=4+211

211 is the residue of 71*6^4 mod 215

in particular 92020 is congruent to 288 mod 323 and mod 284, infact 92020=71*6^4+4

I think that also 331259 has something to do with 71*6^6

so multiple of 43, that is 215, 69660, 92020, 541456 are either of the form 323k+108, or 323k-108, or 323k-288

I notice that (323-108)=6^3-1 and (323-288)=6^2-1

211 and 284 are also the residues of 71*6^4 and 71*6^6 mod 323

the 18-th pg prime is pg(1323),
the 36-th is pg(360787)

modulo (18^2-1=323), 1323 is 31 and 360787 is 319

the difference between 319 and 31 is again 288

1323 is congruent to 31 mod (6^4-4)
360787 is congruent to 6^2*2^3+31 mod (6^4-4)

319 is congruent to 31 mod 6^2 infact

here:

215, 69660, 541456 are congruent to plus or minus 108 mod 323
92020 is congruent to (6^4-1008=288) mod 323

so multiple of 43 are either congruent to plus or minus (18+90) mod (18^2-1) or to 6^4-(18+90+900) mod (18^2-1)

92020 is congruent to (11*6^2-108) mod (18^2-1)
and to - 11*6^2 mod (304^2)

so great fact 215, 69660, 92020, 541456 are either congruent to plus or minus (108*2-1) mod (108*3-1) or to plus/minus (108/3-1) mod (108*3-1)

pg(331259) and pg(92020) are probable primes.

331259=92020+239239 as said

331259 and 92020 are congruent to 5 mod 239
331259 and 92020 are congruent to 6 mod 13

using chinese remainder theorem it yields
something like

239x(1)+13y(1)=1

using Euclidean algor you have the solution y(1)=92

92 are the first two digits of 92020

now 331259 can be rewritten as 239*10^3+92*10^3+331-71-1

331259 and 92020 have the property that they are congruent to the last two decimal digits modulo 9200

331259 is congruent to 59 mod 9200
92020 is congruent to 20 mod 9200

(239*77+1)*5=92020

239*77+1 is a multiple of 215

541456 is congruento to 3/2*(239*77+1) mod (239*215)

331259 is congruent to 5 mod (239*77) and also 92020

(541456-9202*3)/215/239=10

(13*331-1)*7*11+5-7*11*13*239=92020

541456 is congruent to (9203+(71*6^4-1)/5) (mod (239*5*43))

9203 is prime

(541456-9203-(71*6^4-1)/5)/215/239=10

331259-13*(9202*2-1)=92020

92020-13*1720=69660

lcm(215,344,559)=13*1720

67*(16683/67+22360/43-1)=51456

where 16683=9202*2-1-1720 and 22360=lcm(215,344,559)

331259 has the representation

331259=36*(7/180+(92020-4)/10)

Let be Floor(x) the floor function floor(5.5)=5 for example

Floor(239*(331259/(71*6^4))/4)=215

So i suspect that there are infinitely many pg(k) primes with k multiple either of 215 or multiple of something transformed by the Floor function in 215 and in these Cases k is alway 6 mod 13

215=(239*6^2-2^2)/40=(239*6^2-2^2)/(6^2+2^2)

331259=(3*71*6^5+7)/5

71*6^6-(3^3*71*6^5-7)/5=331259

(3^3*71*6^5-7)/5=(10^4-3*331)*331

so we have 92020=71*6^6-(3^3*71*6^5-7)/5-13*(71*6^4-1)/5

(69660/3-860)/13=1720

I see where 215 comes from

(331259-23666)/(12^2-1)=9*239-1=215*10

23666 I used CRT x is congruent to 6 mod 13 and to 5 mod (239*7)

As you can see Both 331259 and 92020 are congruent to 23666 mod 143

But this simply means that both 331259 and 92020 are congruent to 71 mod 143

By the way also 215 is congruent to -71 mod 143

so 215, 92020, 331259 are congruent to plus or minus 71 mod 143

215 is 2 mod (71)
92020 is 4 mod (71*6^3)
331259 is 9203 (prime) mod (71*6^3)

there is clearly a pattern

note that also 541456 is congruent to -(9203*3) mod (7*71)

i have no idea how to develop these ideas but i strongly suspect that there is a structure

It should be clear that

541456 215 69660 331259 92020 are congruent to plus or minus (19+13s) mod 143 for some non negative s.

331259 is also congruent to sqrt(239*9-215) mod 71

215*9=44^2-1

So 331259 is congruent to sqrt(215*9+1) mod 71

another way to see the same thing is that 215 92020 and 331259 are congruent to plus or minus 72 (mod 143)

and 92020 and 331259 are even congruent to - 72 (mod 143*7)

infact 92020+72=92092 and 331259+72=331331

It's easy to see that 9203 conguent to 44 mod 71 and 331259 is as well congruent to 44 mod 71

and 9203 has the same residue 259 mod 344 and mod 559

((541456/921)-((41*43*10+7)/30))^(-1)-7=9203

note that 921=3*307

9210=30*307=((541456/921)-((41*43*10+7)/30))^(-1)

2150=307*7+1

92020*(5879)/999-71.107107107=541456

or 92020*(1763*10+7)/2997-71.107107...=541456

541456-(92020*(1763*10+7)/58790)=10*215*239

331259 is congruent to (71*6^3+307*10)/2 (mod (71*7*6^4))

9203=(331259*2-7*71*6^4)/2

331259=(71*6^3*43)/2+307*5

331259=(2^10+1)*(18^2-1)+184

I would conjecture that if pg(k) is prime and k is multiple of 43,

then k is a multiple of (72+143s) for some positive integer s

the multiple of 215=72+143 are 215 itself, 69660, 92020

the other multiple of 43 is 541456 which is a multiple of 787=72+143*5

multiple of 43 are 215, 69660, 92020, 541456

they are either congruent to (plus/minus) 215 mod 323 or to 288 mod 323

215 and 288 are integers for which a integer solution exists for the equation x^2+71*y^2=z (x,y,z positive integers)

multiple of 43 are 215 69660 92020 541456

Now consider the equations:

x^2+71*y^2=215
x^2+71*y^2=69660
x^2+71*y^2=92020
x^2+71*y^2=541456

there are Always non zero integer solutions x and y

541456 is about 271*999*2 i wonder Why???

so multiple of 86, 92020 69660 and 541456 are congruent to (271*999-1) mod 172

this implies (92020 69660 541456 are also 6 mod 13) that 92020 69660 and 541456 are congruent to 344 mod 2236

and I remember that 92020=69660+22360

this could suggest why multiple of 43 are congruent to plus or minus 344 mod 559

infact (271*999-1) is congruent to 172 mod 559

(271*999-1) is congruent to -172 mod 215

pg(51456) and pg(541456) are probable primes

observing that 51456 is congruent to 508 mod 542 and to 507 mod 999 using CRT:

solutions are 51456+541458n

infact lcm(542,999)=541458=541456+2

331259= 11*2^10+39*2^13+507

51456*9 is congruent to 700^2 mod (164^2)

541456=51456+700^2

(700^2-164^2)/51456=3^2

(10*700^2-164^2)/3^2=541456

So i wonder inf there are infinitely many primes pg(k) with k multiple of 86 with the property that k is congruent to 164^2 mod (268) as pg(541456) and pg(92020)
More generally if k is multiple of 86 and k is multiple of 3, 69660 is the example

k/3 is congruent to - 164^2 mod 268

if k is not multiple of 3 and k is multiple of 86, then

Examples are 92020 and 541456 are congruent to 164^2 mod 268

541456=271*9*999*2-18

271*9=2439

Maybe this explain the fact that 541456/41 and 51456/41 have a repenting term 2439 and so they are 10^m mod (700^2+239)*2^8/(271*9)=51456

(2439*111-1)*2-(2439*201-239)=51456

From above reasoning it results that

541456 and 51456 are both congruent to -239 mod 245=7*25

In subastance here the formulas

541456=245*2211-239
51456=245*211-239

Where 211 is prime and 2211 is a multiple of 67 as 51456

(541456-92020) is a multiple of 559 and 67

(541456-92020)/(67*3)=2236

69660+22360=92020

But ed can substitute 67*3 with 2211/11

So (541456-92020)*11/2211=2236

i note also that (541456-210*1001+13)=331259

92020 is so congruent to (245*2211-239) mod (3*67*43*13)

and (245*2211-239=541456) is congruent to (331259+13) mod 559

i think that it shuould be possible to prove that when pg(k) is prime and k is congruent to 6 mod 13, (examples known 215, 69660, 92020, 331259, 541456) then k is congruent to plus or minus (344-13s) mod 559
with s=1 in the case of 331259 which infact is congruent to 331 mod 559 and s=0 in all the other cases where k is a multiple of 43

i think that something interesting could be found examining this formula:

f(x)=((71*6^x+4)/10)*6^2-13 for x integer

x=4 f(4)=331259
331259 congruent to -13 mod 9202

i think that the study of this function could shed light on these numbers

331259 is congruent for example to -7 mod 55211=f(3)

331259=92020+239239

note that also 239239 is congruent to -13 mod 9202

239239 is congruent to 546 mod 559

215*(239239-546)/559+215=92020

I suspect that 546 is not a random residue

infact there are primes pg(k) with k multiple of 546

for example pg(75894) is prime and 75894 is multiple of 546

modulo 559 75894 is 429 (92020=(429^2-1)/2)

(239239-546)/559+2=429

so( ((239239-546)/559+2)^2-1)/2=92020

(239239+(12^2-1)*4)=429

Given the attention this question has received, it is disappointing that it is closed. I have a more concise answer that I cannot post other than as a comment: 245⋅22…11−239=245(2000*(10^m−1)/9+211)−239. Using the congruences: 245≡−1, 2000≡−9, 451≡0mod41 we get (−1)(−9)10m−19+(−1)(211)−239≡10m−1−211−239=10m−451≡10m

245 is congruent to -1 mod 41

modulo 41 we have 245*211=-211

-239-211-1 =451 congruent to 10^m mod 41

pg(451) is prime

((71*6^4+4)-344)/559=164

but we saw before that

51456=(700^2-164^2)/9
541456=(10*700^2-164^2)/9

so we arrived to this:

(700^2-(((71*6^4+4)-344)/559)^2)/9=51456
(10*700^2-(((71*6^4+4)-344)/559)^2)/9=541456

So multiple of 43

215 69660 92020 and 541456 are of the form

s(71*6^4-340)/52 +r with r being the residue 1763s+r such that is congruent to 10^m mod 41

340-52=288 maybe this explain why 92020 is congruent to 288 mod 323???

I think yes because 92020=52*1763+344

(71*6^4+4)
is congruent to 344 both modulo 1763 and modulo 559

(92020-344)/559+(92020-344)/1763=6^3

92020=(71*6^4+4)
331259=(71*6^6+4+10)/10

71*6^6 is congruent to -12^2 mod (71*6^4+4)

2236=(331*1001-81*143)/143

Anyway we have a fact:

92020 and 331259 are congruent to 71*3^3 mod (239*13)..so I wonder if there are infinitely many of such exponents

By the way I note that

541456 is congruent to 71*3^3 mod (1001)

541456 is congruent to 331*1001+71*3^3 mod (1001*13)

Or 541456 is congruent to 71*3^3 mod (11*1001)

so 541456=71*3^3+11011n where 11011 is 3^3 in base 2

71*3^3 is congruent to 344 mod 143

I think that this could be useful

It holds 541456-(331259)+13-210*1001=0

So this could be useful if you have in mind that 331259 is congruent to 71*3^3 mod (239*13)

92020=541456-13*(449*77-1)
331259=92020+239*1001

it is clear that there is a relationship

541456 is congruent to (331259+71*3^3+111) mod (239*13)

331259=92020 mod(239*13)

so 541456 is congruent to 2*71*3^3+111 mod (239*13)

92020 and 331259 are congruent to 71*3^3 mod (239*13) and to 71*3^2+13 mod (1001)

92020 and 331259 are congruent to 929 mod (1001)
541456 is congruent to (929-13) mod 1001

69660*559 is congruent to (331259-559-331) mod (1001)

69660*559 is so also congruent to (92020-559-331) mod (1001)

so 69660*559 is either congruent to (x-890) mod 1001 or to (x-903) mod (1001)

where x=541456, 331259, 92020 numbers congruent to 6 mod 13

331259 and 92020 are -72 (mod 1001) and so 69660*559 is congruent to (x-890) mod (1001) with x=331259, 92020
if x=541456 not congruent to - 72 mod 1001, then

69660*559 is congruent to (x-903) mod 1001

but this is equivalent to say

69660*559 is congruent to either (x +111) or (x+111+13) mod (1001)

x=331259, 541456, 92020

541456, 331259, 92020 are congruent to either (13*2-111) or to (13*3-111) mod 1001

69660 is congruent to 591 mod 1001

591*559 is congruent to 39 mod 1001

69660/3 is congruent to (14^2+1) mod 1001

92020 and 331259 are congruent to -(14^2-111) mod 1001
541456 is congruent to -(14^2-111-13) mod 1001

so

69660 is congruent to 591 mod (1001)
331259 is congruent to 559*591-111 mod (1001)
541456 is congruent to 559*591-111-13 mod (1001)

92020 the same as 331259 if i am not wrong

so 69660 is (14^2+1)*3 mod 1001

331259 is congruent to 559*3*(14^2+1)-111 mod 1001...

-(69660/81) is congruent to 141 mod 1001

so

541456 is congruent to (4*(14^2+1)-860-13) mod (1001)

92020 and 331259 to (4*(14^2+1)-860) mod 1001

69660 is congruent to 3*(14^2+1) mod 1001

14^2+1=197 is a prime

69660/81=860

541456 is congruent to 3*860-41*5*13 mod (1001)
331259 and 92020 are congruent to 3*860-41*5*13+13 mod (1001)
69660 is congruent to -2*41*5 mod 1001

541456 is congruent to 344-13*2^5-13 mod 1001
331259 and 92020 are congruent to 344-13*2^5 mod 1001

541456 is congruent to (344+13*2^5-13*2^6-13) mod 1001
92020 and 331259 to (344+13*2^5-13*2^6) mod 1001
69660 to (344+13*2^5+13*2^6) mod 1001

-(344+3*13*2^5) is congruent to 410 mod 1001
-(344-13*2^5) is congruent to 72 mod 1001
-(344-13*2^5-13) is congruent to 85 mod 1001

69660 is -410 mod 1001
331259 and 92020 are -72 mod 1001
541456 is -85 mod 1001

pg(2131) pg(2131*9=19179) and pg(92020) are probable primes

2131 is congruent to -1^2 mod 164
19179 is congruent to - 3^2 mod 164
92020 is congruent to 4^2 mod 164

92020 is congruent to 43*2^3 mod (2132) and congruent to 43*3^2 mod 2131

19179 is congruent to -81 mod (2131+9)
2131 is congruent to -9 mod (2131+9)
92020 is a multiple of (2131+9)

This implies that

(92020-2131) is congruent to 9 mod 4280
(92020-2131*9) is congruent to 81 mod 4280

92020 is congruent to 2140 mod 4280

92020 s congruent to 2140 mod 8988

2131=p is prime
pg(2131) is prime and pg(2131*9=19179) is prime

pg(92020) is prime

pg(69660) is prime
92020=69660+2236*10

92020 can be written both as (2236*43+344) and (2132*41+344)

so 92020 leaves the same residue 344 mod (41*2236) and mod (2132*43) so 92020 is also congruent to 344 mod 1763

Also 541456 and 69660 multiple of 86 are 344 mod 2236

So the exponents multiple of 86 are 69660 92020 and 541456

They are congruent to 344 mod 2236

But only 92020 is congruent to 344 modulo 2236, mod 1763 and modulo 2132=p+1=2131+1 where this prime p 2131 gives the other two probable prime pg(2131) and pg(19179)

19179 is congruent to -214 mod 1763

92020 is a multiple of 214

-2131*3^2*430 is congruent to 344 mod 1763

92020=214*430

so

modulo 1763

-2131*3^2=214

92020=214*430 congruent to 344 mod 1763

pg(331259) is prime
331259 is congruent to 6 mod 13

331259 is congruent to 344-559*3 mod (2132)

pg(56238) and pg(331259) are probable primes

the difference 331259-56238 is congruent to - 7 mod (2132*43)

331259 is -7 mod 13, 56238 is 0 mod 13

331259+7-56238=43*2132*3

this is equivalent to 92020+239*1001+7-56238=43*2132*3

92020+239239=331259

239239 is congruent to (56238-344-7) mod (2132*43)

92020 is congruent to 344 mod (2132*43)

331259 is congruent to (56238-7) mod (2132*43)

92020 is congruent to 344 mod 2132

331259 is congruent to 344*2+111 mod 2132

56238-7 is congruent to 344*2+111 mod 2132

pg(451=11*41) is prime pg(2131) is prime pg(92020) is prime

92020 is congruent to -2131*9*430 both mod 1763 and mod 451 -2131*9 is 214 mod 451 and mod 1763

92020 is congruent to -215 mod (6149=11*43*41)

541456 is congruent to 344 mod 6149

(215+2131*9-1)/43=451

11*43*41 mod 2131 is 214

92020=214*430

Mod 2131

214*430 is 387

So 92020 is congruent to 387 mod 2131

-(1763*11) mod 2131 is 71*3^3

331259 and 92020 have something tondo with 71*3^3

In fact 92020 and 331259 are congruent to 71*3^3 mod (239*13)

71*3^3=2131*10-41*43*11

This means that 331259 and 92020 are congruent to (2131*10-451*43) mod (239*13)

and to (21310-451*43+13) mod (1001)

541456 is congruent to (21310-451*43) mod (1001)

Consider this set of congruences:

x==-13 mod 214
x==6 mod 13
x+13=344 mod 43

Using CRT calulator I found the solution

x=92007+119626k

92007+13=92020 and pg(92020) is prime

92007+119626*2=331259 and pg(331259) is prime

another curio:

pg(1323) is prime

pg((1323*10+3)*3=39699) is prime

1323 is congruent to -215*3 mod 984

39699 is congruent to -215*3 mod 984

Pg(69660) is prime

69660 is multiple of 215*3=645

1323/3=441

69660 is congruent to -215*3+441 mod 984

Or to -215*3+21^2 mod 984

modulo 984, the number 69660 is a perfect square (42*9)^2
modulo 984

69660 is -1323*108=-(42*9)^2

as you can see 69660 is -1323*108=-(42*9)^2 mod (6^3)

now i consider this modular equation

1323x is congruent to -6^3 mod 984

the soluton wolphram gives me is (40+328n)

for n=-1 you get -288

92020 is congruent to 288 mod 323
and 215, 69660 and 541456 are + or - 108 mod 323

modulo 328:

21^2 is -215

-69660=21^2*42^2 modulo 328

-92020=107*42^2 modulo 328

-215 and -69660 in a certain sense are perfect squares (21^2 and 378^2)

378^2 is congruent to 21^2 mod (42*323)

92020=428*215

428 is congruent to -13 mod 21^2

92020 is congruent to 215*10^2 mod (215*328)

Ah ah this is weird

Also 92020 is a perfect square

-210^2 mod 328

Infact 428 is 10^ 2 mod 328

So mod 328 92020 is -210^2

1323 and 39669 are 11 mod 328

69660 modulo 328 is 11*108

Modulo 328 92020 is 11*4*(108^2-1)

-69660 is congruent to 204 mod 984

204=645-21^2

69660 is a multiple of 645

1323=42^2-21^2

1323*108 mod 984 is 204 as -69660 mod 984

69660 is congruent to 21^2-645 mod 984
92020 is congruent to 13^2-645 mod 984

1323 is 11 mod 328
39699 is 11 mod 328
69660 is (11+21^2) mod 328

1323 is 339 mod 984
39699 is 339 mod 984
69660 is (339+21^2) mod 984

69660 mod 984 is 780

42^2 also is 780 mod 984

this because
1323 is congruent to 339 mod 984

69660 is congruent to (339+21^2) mod 984

so
69660 is congruent to (3*21^2+21^2=42^2) mod 984

239*7*11 is -1 mod 172

69660 92020 and 541456 are congruent to (239*7*11-171) mod 2236

I notice that 92020 and 331259 are both congruent to 5 mod (239*7*11)

331259 is congruent to (239*7*11-171-13) mod 2236

modulo 2236 infact

239*7*11-171 is 344

239*7*11 is 515 mod 2236

so

239*7*11+1 so is 516 mod 2236

69660 is divisible by 516

92020, 541456 are 172 mod 516

331259 is -13 mod 516

-69660 is congruent to 1323 mod (239*11)
69660 is congruent to 6 mod 13
using wolphram alpha solution is
:
69660=1306+34177*2

92020=1306+22360+34177*2
331259=1306+22360+34177*9

what is quite clear is that

there are k such that pg(k) is prime with k congruent to 6 mod 13 and k of the form 215+1001s-287 (case 92020 and 331259) and 541456 which is of the form 215+1001s-287-13

so k is of the form -72+1001s or -85+1001s

215 is congruent to -(929-11*13) mod (7*11*13)
92020 is congruent to (929) mod (7*11*13)
331259 is congruent to (929) mod (7*11*13)
541456 is congruent to (929-13) mod (7*11*13)

929 and 331259 are primes congruent to -72 mod (7*11*13)

so mod 143

215 331259 92020 ...are plus or minus 71 mod 143
541456 is (71-13=58) mod 143..

215 is congruent to -786 mod (11*13*7)

786 is a number of the form 71+143*s

331259 and 92020 are congruent to 929 mod (11*13*7)

929 is of the form 71+143s

so these exponents leading to a prime with k congruent to 6 mod 13 have this property

215 is congruent to (71+143s) mod 1001
541456 is congruent to (58+143s) mod 1001
331259 is congruent to (71+143s) mod 1001
92020 also
I dont know 69660??? maybe not but 69660 is the only one multiple of 3

i note that 215 541456 331259 92020 69660 are congruent to plus or minus (71+r) mod 1001

where r is a 13-smooth number which is not a 11 smooth number

r infact can be one of these numbers: 845, 858 or 520

520=71+18^2+14^2
845=71+18^2+14^2+18^2+1
858=71+18^2+14^2+18^2+14

multiple of 43 (215, 69660, 92020, 541456) are congruent either to phi(323)=288 mod 323 or to +/- phi(324)=108 mod 323

phi is the Euler function

215-eulerphi(324)=107

92020 is a multiple of 107, 69660 is 3 mod 107...the thing becomes too difficult

multiple of 215=6^3-1 are 69660 and 92020

69660=324*215
92020=428*215

324 and 428 are numbers n such that

n-eulerphi(n)=6^3

is it a chance that the other multiple of 43 (and not multiple of 215), that is 541456 is congruent to 6^2 mod 428?

Another way to see the problem :

215 69660 92020 541456 are congruent to plus or minus K mod 323

Where K is either 215 or 288

215 and 288 are numbers of the form 41s+r where r is in the set (1,10,16,18,37)

So 215 69660 92020 541456 are of the form 41a+b (b in the set 1,10,16,18,37)and congruent to +- (41s+r) mod 323

215 and 288 are numbers of the form (3+n)*71+2^(n+1) for n non negative integer

note that 12345679*323 divides (10^288-1)

12345679*323*81 divides (10^(phi(323)-1)

phi(323)=288

92020/324 gives a repeating decimal period of 012345679

92020 is congruent to 288 mod 323

215, 69660, 541456 are of the form k*(41s+r)*323 + or minus 215, where r is in the set (1,10,16,18,37) and k some nonnegative integer

541456=323*1677-215
215=0*(41s+r)+215
69660=323*215-215

the only exception is 92020 which is congruent to 16 mod 41

92020 is of the form 323*284+phi(323)

now pg(1323) and pg(39699) are probable primes

1323=441*3
39699=4411*3^2

numbers of the form 441...4411...are

(397*10^n-1)/9

1323 and 39699 are congruent to -645 mod (41*3*2^3)

(397*10^n-1) is divisible by 9209 for a certain n...

397*10^5-((397*10-1)/9-126)*1001-5^3 is a multiple of 92020

39699 and 1323 are of the form 328n^2+11

maybe there are infinitely many of such exponents

541456 is a multiple of 787

69660 is a multple od (787-13=774)

541456-(222*2021+787-13)=92020

331259-118*2021-(787-13*2)=92020

from the equation above

541456-(222*2021+787-13)=92020

if we work mod 787

0+715+13=728

so 92020 is congruent to (3^6-1) mod 787

92020 is also congruent to 43*2^n mod (787-13)

92020-69660=22360 is also congruent to 43*2^n=688 mod (787-13)

69660 is congruent to (3^6-1-18^2) mod 787

22360 is congruent to 18^2 mod 787

92020=69660+22360

92020 is congruent to 3^6-1 mod 787

69660 is divisible by 18^2 and it is of the form (2^j*3^k)-18^2 with (2^j*3^k) congruent to 3^6-1 mod 787

69660 is also congruent to 18^2 mod (107*2)

92020 instead is a multiple of (107*2)

92020 is a multiple of 428

69660 is congruent to 18^2 mod 428

541456 is 0 mod 787

curious that pg(787-428=359) is prime

215 69660 92020 541456 are +/- 344 mod 559

lcm(215,559,344)=22360

69660+22360=92020

22360 is of the form 787n+324

maybe working in the field F(787):

the inverse mod 787 of 107 is 559

92020 is congruent to (107*73) mod (107*787)

541456=107*2942+787*288+6

541456=107*(17*19*10-phi(17*19))+787*phi(17*19)+6

-215 (because odd so signus -), 69660, 92020, 541456=G are solutions of the diophantine equation:

107x+787y+6=G

I think that we think that the inverse mod 239 of 428 is 43...

43*428-1=7*11*239

and that lcm(428,559)=239*7*11*13+13

then maybe we could explain why

541456+(72)+13 is congruent to 0 mod 1001

331259+72 is congruent to 0 mod 1001=7*11*13

92020+72 is congruent to 0 mod 1001
...

A strategy could be working in the field F(239)

107*172 is congruent to 1 mod (239*7*11)

so the inverse of 172 mod 239 is 107

172=344/2

215 69660 92020 and 541456 are all congruent to +-344 mod 559

92020 is a multiple of 107

331259=92020+239239

69660 92020 and 541456 are multiple of 172...but 172 in the field 239 is (107^(-1))

multiple of 86 are 69660 92020 541456...

69660*(1-1/172) is congruent to 0 mod (3^2*5*19)
92020*(1-1/172) is congruent to 0 mod (3^2*5*19=855)
541456*(1-1/172) is congruent to 513=19*3^3 mod (3^2*5*19)

855 is congruent to -1 mod 107

69660*(1-1/172) is congruent to -9^2 mod 107
92020*(1-1/172) is congruent to 0^2 mod 107
541456*(1-1/172) is congruent to -3^2 mod 107

-2^9-7^3 is congruent to 6^3-1 mod 107

-2^9=-19*3^3+1

6^3+7^3 is the famous 559

344=7^3+1

lcm((6^3+7^3),(7^3+1),(6^3-1))=22360

69660+22360=92020!

curious that 559 is congruent to (1-2^9) mod 107

and to (1+2^9) mod 67

maybe this is not chance because there are primes pg(67s) like pg(51456) but i dont know

92020 for example is congruent to (6^3+2^9) mod 787
541456 is a multiple of 787
69660 is congruent to (6^3+2^9-18^2) mod 787

6^3+2^9=3^6-1

so 3^6 congruent to 2-7^3 mod 107

541456=344*1574=-2^9*1574 mod 107

2^9 mod 107 is 84

1574 mod 107 is 76

so 541456=-84*76 mod 107

but incredibly

541456=-84*76 mod (107*10*2^9)

and the difference 541456-107*10*2^9 is divisible by 456

in other words:

541456=(6^3+7^3+2^9-1)*2^9-84*76

if you reduce the right side mod 107

6^3=2
7^3=22
2^9-1=83

2+22+83=107

69660 is congruent to (6^3+1)*(6^3-1) mod (5*43*107)

infact 324 is congruent to (6^3+1) congruent to 3 mod 107

69660=324*215

so 69660 is congruent to (6^6-1) congruent to 3 mod 107

69660 is congruent to (6^6-1) mod (215*107)

215*107 is congruent to 1 mod (71*6^2)

is this connected to the fact that for example 92020 is about 71*6^4???

69660-(6^6-1) divides 92020

92020=69660+22360

69660 is a multiple of 645

22360=215*107-215*3

69660 is congruent to 645 mod (215*107)
69660 is congruent to -22360 mod (215*107)

using wolphram (chinese remainder theorem) this leads to numbers of the form:

645+23005n

6^6-1 and 69660 are so nummbers of the form 645+23005n

92020 is divisible by 23005

pg(331259) is probable prime

331259 is congruent to 9203 (prime of the for 107n+1) mod (107*215-1)

((215*107-1)*12^2+12^2)/10-(239239+13)=92020

and 541456 is congruent to the curious 12341 mod (215*107), 12341=7*41*43, this remainds me the famous 1763

69660 is congruent to 6^6 mod (71*6) congruent to 222 mod (71*6)

69660/3-222/3-786=22360

a rapid calculation from above leads

69660+1 congruent to 2^6*(6^3+1)+2^15 mod (215*107)...
this yields 69660+1 is congruent to (13888+9763) mod (215*107)

69661-13888-9763=46010

46010 is a divisor of 92020

69660=3^2*(71^2+71*36+71*2+1)

so this leads to a polynomial x^2+36x+2x+1

71+6^2=107 so I arrived to

69660=3^2*(71^2+71*36+71*2+1)

feom this I found this polynomial:

(x+19)^2-9^2*10^2=0

whcih has solutions:

x=71

x=-109

71*109+1=6^2*(6^3-1)

from this we obtain

(9^2*10^2-19^2+1)=6^2*(6^3-1)=7740, which divides 69660

69660 has also the factorization 6^2*(44+1)*(44-1)

pg(331259) is probable prime

331259 is congruent to (71*18^2-1)=215*107-2 mod (456) and mod (26^2)

maybe there is a link 513=2^9+1=19*3^3 and 456=2^3*3*19, the difference is 15*19 congruent to 1 mod 71

215*107-15*19+6^2*(6^3-1)-90^2=22360

where 215*107 15*19 and 6^2*(6^3-1) are 1 mod 71

pg(331259) is probable prime

331259=(13*(71*11*18^2+1)+71*18^2+1)/10

71*18^2+1=23005 which divides 92020

69660-71*18^2=6^6

and pg(69660) is prime

(13*(71*11*18^2+1)+71*18^2+1) is congruent to -13*10 mod (215*107)

maybe it has something to do with the fact that

23005*143 is congruent to 1 mod 71

infact 23005 is 1 mod 71 and so 23005*143 is 1 mod 71

I don't know how to connect things but

107*73 is congruent to 1 mod 71

and so all the numbers of the form

(73+71s)*107 are congruent to 1 mod 71...

numbers of the form 73+71s are 215 and 428

maybe it is not a chance that there 92020 is divisible by 428 (and also 215)

92020 is also divisible by 23005 which is of the form 73+71s+1

215 and 428 in the field Z 71 have 36=6^2 as inverse

maybe it is not chance that 541456 is 36 mod 107

and 92020 is (324-36=288) mod 323

pg(359) is prime

359-71=288 the famous 288=phi(323)

pg(359) comes after pg(215) which is probable prime

359 and 215 are numbers of the form 71+144s

69660=71*324+6^6 if we rearrange

324*(71+144)=69660

I note that 359=19^2-2

pg(359) is prime

215, 69660, 92020, 541456 are congruent to plus or minus (19^2-73s) mod 323

infact 215=19^2-73*2 and 288=19^2-73

pg(51456) is prime

2^4*(73*49-361)=51456

pg(67) is prime

215 is a number of the form 67+2+73s

also 288 is a number of the form 67+2+73s

maybe it is not a chance that also 1456 the final digits of 51456 and 541456 is of the form 69+73s

(1456-69-1)=1386

1386*239+5=331259 and pg(331259) is prime
(1386-1001)*239+5=92020 and pg(92020) is prime

215 288 and 1456 in the field Z 73 have 18 as inverse...curious that 1456*18-1=73*359 and pg(359) is prime

by the way 541456 is congruent to 1456 mod 288

108 and 288 are numbers of the form 54+18s

215 69660 92020 541456 are congruent to +/- (108 or 288) mod (18^2-1)

also 774 is a number of the form 54+18s and 774 divides 69660

69660 is divisible by 18^2 which is a number of the form 54+18s, by 774 which is again a number of the form 54+18s, by 90 which is a number of the form 54+18s and by 108 which is of the form 54+18s

288 and 108 are two smooth numbers congruent to 18 mod 90

71*18+224^2+700^2+2=541456
71*18+224^2+2=51456

224^2+700^2 is congruent to 215*2=43*5*2 mod (71*18)

224^2=(67*3-5)*2^8

51456=67*2^8*3

curious that 224^2+1 and 700^2+1 are both prime

69660=(9*71+6^4)*6^2

645 divides 69660

92020=69660+22360

22360=23005-645

23005=215*107 divides 92020

23005 is congruent to 1 mod 71

curious that there are pg(k) primes with k multiple of the reverse of 645=546

645=(18^2-1)*2-1

92020=215*428

Both 215 and 428 are congruent to 2 mod 71

The inverse of 215 in the ring F426 is 107

curious that pg(7) is prime , 7 is prime...the inverse of 7 in the ring F426 is 61...(61*7-1)/426=1

541456 is a multiple of 787

in the ring F426 the inverse is 367

367*787=288828

288828=6^2*(8*1003-1)

(359*89-1)/426=75=(8*1003+1)/107

8*1003 is 1 mod 71

pg(359) is prime and the inverse mod 426 of 359 is 89

remark:

in the ring F426

a^2 is congruent to 1 mod 426, for a=143

(143*143-1)/426=288

lcm(426,288)=143^2-1

maybe there is a link to the fact that

541456-429*10^3-143^2+13=92020

in the ring Z428

(215*215-1)/428=108

(211*71-1)/428=35

215 69660 92020 and 541456 are congruent to + or - (108 or 35) mod 323

pg(3371) is probable prime with 3371 prime
pg(331259) is proabble prime with 331259 prime

3371=59+46*72
331259=59+46*10^2*72

I would conjecture that there are infinitely many prime pg(k) with k a prime of the form 59+72s.

I note that 46*10^2+1=4601 which divides 92020

this is equivalent to primes of the form (72b-13), as Peter Scholze pointed out

I note that 331259 equiv (congruent) 999 mod (3371-1)

3371 and 331259 leave also the same resiude 13 mod 46

so they are primes of the form 59+1656x

1656 is a multiple of 46 naturally

I think it is not a chance that 4601 divides 92020

72*4601-13 is 331259

so 3371 and 331259 are primes of the form 72*k-13 with k congruent to 1 mod 46

because 331259=239239+92020 follows that

(239239+13) is a multiple of 428 and (239239+13)/428=the famous 559

3371 and 331259 are primes of the form 59+3312x

331259=(4601*4-1)*13+4601*20 where 4601*20=92020

because 59 and 3312 are coprime we expect infinitely many primes of the form 59+3312x so there could be infinitely many pg(59+3312x) primes with 59+3312x prime

I note that 3371, 331259, 92020 are all congruent to 5 mod 11

so 3371 and 331259 are primes of the form -589+792s

curious that 589*792+1=683^2

putting all together we have that

3371 and 331259 are primes of the form -14845+18216s.

infact 3371 and 331259 are 13 mod 46, -13 mod 72 and 5 mod 11

3371 and 331259 leave the same remainder 82 (mod 23*11)

so they are also primes of the form 23*11*s+82

3371=82+253*(13)
331259=82+253*(13+6^4)

3371 and 331259 are primes of the form 59+414s where 414 is 1 mod 59 and 0 mod 23

4601 which divides 92020 is 1 mod 23 and -1 mod 59

so 3371 and 331259 are primes congruent to 59 mod (46*72)

3371 and 331259 are congruent to 13 mod 46

(3371-13)/46=73 which is congruent to 1 mod 72

(331259-13)/46=19*379 which is congruent to 1 mod 72

19*379 is congruent to 51^2 mod 46

51^2 is congruent to 1 mod 26

4601 which divides 92020 is congruent to -1 mod 26

19*379+1 is a multiple of 26

4602 is a multiple of 26 and 59

19*379+1-4602+1=51^2

51^2 is congruent to 1 mod 26
51^2 is congruent to 2 mod 23

4601 is congruent to 1 mod 200 4601 divides 92020

19*379 is congruent to 1 mod 200

we can say

3371 and 331259 are congruent to 13 mod 46

3371 and 331259 are congruent to -13 mod ((46s+1)*72)

3371 infact is congruent to -13 mod (47*72)
331259 is congruent to -13 mod 4601*72 where 4601=46s+1

92020 is 0 mod 4601

72 is the least integer such that

4601*x is congruent to 1 mod 337

3371 is 1 mod 337

69660 (divisible by 215*3) is congruent to 215*3 mod 4601

4601 is congruent to -1 mod 59

22360 is congruent to -1 mod 59

4601 divides 92020

69660+22360=92020

774 divides 69660, 428 divides 92020

331259 is congruent to -13 mod (774*428)

774*428 is congruent to 1 mod 337

3371 is 1 mod 337

91*100 is congruent to 3371 mod 337

331259 is -12 mod 337 and -13 mod (428*774)

331259 and 3371 are primes of the form 59+3312s

3312 is congruent to 2 mod 331

92020=428*215 is 2 mod 331

428*774 is congruent to -59 mod 331

curious that pg(1323) is prime and 1323 is congruent to -1 mod 331

69660 is (6^4+4) mod 230

so 69660 is 200 mod 230

the inverse mod 230 of 43 is 107

so 5*324 is 10 mod 230

5*94 is 10 mod 230

5*47*2=5*94

the inverse mod 230 of 47 is 93

so 10 is congruent to 930 mod 230

92020 is congruent to 20 mod 920

43*107*72 is congruent to 1 mod 337

43*107*72 is congruent to 3371 mod 337

43*107*72-1=331271, the number is 3371 with a 12 sandwiched 33(12)71

the difference between 331271 and 3371 is 1093*300 where 1093 is a Wieferich prime

300 is the least integer such that 1093x is congruent to -1 mod 337

331259 is congruent to -12 mod 331271 and to -13 mod (43*107*72)

3371 is congruent to -1001 mod 1093
331259 is congruent to -1001-12 mod 1093

working mod (-1093)

3371 is -1001

331259=3371+327888k

working mod (-1093)

331259=-1001-12s

3371 is congruent to -12*28 mod 337

the inverse mod 337 of 28 is 325

3371*325 is congruent to -12 mod 337

so 3371*325 is congruent to 331259 mod 337

and so 331259 is congruent to -12 mod 337

Se know that 69660 is congruent to 6^6 mod 71

23004 (23005 divides 92020) has the form 288*3^x-18^2

Also 69660 should have the form 288*3^x-18^2

6^6 is 648 mod (71*324)

(69660+648)/(288*3^2-324)=31

69660 is divisible by 540=288*3-324

given 69660 is congruent to 2^3*3^4 mod 213

after some passage :

111 is congruent to 2^7*(2^7+1) mod 213

this implies

324 is congruent to 2^7*(2^7+1) mod 213

and 69660=324*215 so is congruent to 2*(2^7+1)*2^7 mod 213

(2^7+1)=129 is a divisor of 69660

I notice that 128*129=2^7*(2^7+1) is congruent to -1 mod 337

so 3371 is congruent to (128*129+2) mod 337

I notice that 128*129+2 is a multiple of 359 and pg(359) is prime

I notice that 128*129 is congruent to -7^3 mod 3371

3371 is congruent to 336*128*129 mod 337

336*128*129-3371=5544661=10*(2^10-1)*271*2

331259 is congruent to 5544649 congruent to 325 mod 337

so 331259 is congruent to 11*83*6073 mod 337

and so
331259 is congruent to 239*7 mod 337

if we multiply by 143

143*331259 is congruent to 239239 mod 337

331259=92020+239239

because 239239 is -(2^5+1) mod 337, then

331259-92020 is conruent to -(2^5+1) mod 337

92020 is -10 mod 9203 (this strange prime is 1 mod 43*107)

331259 is 9203 mod 23004=215*107-1=71*...
331259 is -7^2 mod 9203

71*6^6+1 is congruent to 2131*1399 mod 9203

pg(2131) is prime

71*6^6 is congruent to 8699 mod 9203

71*6^6 is congruent to 8699 mod (9203*359)

pg(359) is prime

331259 and 71*6^6 are both congruent to -72 mod 331 or (259) mod 331

(71*6^6+72)/331-(331259+72)/331=9007=10^4-3*331

-71*6^4 is congruent to 2 mod 331 and 92020 is congruent to 2 mod 331

in particular (71*6^6+72) is divisible by (92020-2=92018, which is multiple of 331)
(331*139-1)/2=23004=71*...
23005 divides 92020

139*331 is -1 mod (43*107)

curious that 331259 is congruent to -(9203*2^2+1) mod (331*139)
(331*139-215*107)+6^6=69660

331259 is congruent to -2000 mod 9007=10^4-3*331

71*6^6 also is -2000 mod 9007

curiously 92020 is congruent to -(84^2+1) mod 9007

after a post on mathexchange I had a proof that:

6^(6+35j) for some j, has the form 648+23004s

69660 has the form 648+23004s

69660 is divisible by (6^5-6^2)

6^5+1=7777

6^5 is congruent to (6^2+1) mod 71

i didn't realized that

331259 is congruent to 9203 mod (71*324)

331259 and 9203 are primes

331259 and 9203 are both congruent to 44 mod 71

(331259-9203)/71=4665

6^6=46656

4665 has in common with 46656 the first four digits

6^6 is congruent to 44^2 mod 2236

I remember that

69660=(44+1)*(44-1)*2

and 69660 is congruent to 6^6 mod 23004

331259=(2236*2+193)*71+44

193 is 1936=44^2 with a six truncated

mod 2236 331259 is 331

69660=(44^2-1)*36=(193*10+6)*36

(6^6-1) is congruent to 1935 mod 2236

1935=(44^2-1) divides 69660

I looked at the factorization of 215, 69660, 92020, 541456 (multiple of 43 leading to a pg prime)

215=43*5 for example

every prime in the factorization of one of these numbers is a quadratic residue mod 71. This is quite surprising.
for example 43 is a quadratic residue mod 71
also 5
but also 787 which divides 541456

331259 is congruent to (4667) mod 6^6

4667 is a multiple of 359 (pg(359) is prime)

-69660 is congruent to 345 mod 359 as 6^6 is congruent to 345 mod 359

69660+6^6 infact is 359*324=359*18^2

consider

71x is congruent to -1 mod 215

359x is congruent to 1 mod 215

x=109+215n

71*(215+109)=23004
71*109=7739

7739+1 divides 69660
23004+1 divides 92020

curious that (359*109-1)/215=182

and 541456 is congruent to 182 mod 2131, pg(2131) is prime

182^2+1=33125 and pg(331259) is prime

359*(109+215x)-1

for x=2 we have 193500

1935 divides 69660

(359*(109+215x)-1)/10^x is an integer for x=2

69660 is congruent to 6^6 mod (71*324) and to -6^6 mod (359*324)

pg(359) is prime

the difference between 359 and 71 is the famous 288=phi(323)

I think it's not chance that 92020 is congruent to 288 mod 323

288=2*6^6/324

in the ring F116316 -69660 is congruent to 6^6 mod (116316)
in the ring F116316 (116316=359*324)
:
-331259 is congruent to 133^2 mod (116316)
pg(331259) is prime

pg(92020) is prime.

92020=215*428

428=21^2-13

pg(1323) is prime. 1323 is a multiple of 21^2

pg(331259) is prime

331259+13 is multiple of (43*107)

331259+13+6^2 is multiple of 9203

9203 is a prime of the form 96^2-13

69660 and 92020 can be written in the form 3+215x+23004y, for some x,y

92020=3*215^2-6^6+1

(6^6-1) is congruent to 3 mod 107

69660 is congruent to 3 mod 107

92020=69660+22360

-22360 is congruent to -324*213 congruent to 3 congruent to (6^6-1) mod 107

22360*36 congruent to 23004 congruent to -3*36 congruent to -1 mod 107

22360*36 is congruent to -5*36 mod (71*324*35)

Divisors of 69660 are 36 and 7740, Numbers of the form 36+107s. 7740=36+107*72 and is congruent to 1 mod 71

4472*36 congruent to -36 mod 23004

22360=4472*5

4472 is congruent to -1 mod 71

And 4472=lcm(344,559) the famous 344 and 559

pg(331259) is prime

331259 is prime congruent to 6 mod 13

331259 is congruent to (2^8+1=257 prime) mod (4473)

22360=(22622-262)

331259 is congruent to 257 congruent to (22622-262)+262 mod (4473)

subtracting 5 we have

1386*239 congruent to 252 congruent to 359*63 mod (4473)

dividing by 63

22*239 congruent to 4 congruent to 359 mod 71

so 22*239 congruent to 4 congruent to 359 congruent to 92020 mod 71

331259=92020+239239

92020=4601*20

4601 congruent to - 5 mofd 71

69660*63 congruent to -(4601+5)*63 congruent to 9*63 congruent to -1 mod 4473

so

69660*63 congruent to -92020-4601*43-5*63 congruent to -1 mod 4473

so

69660*63 congruent to -92020+76*10^2 mod 4473

11701*5 congruent to 43*324 mod 23004, 43*324 congruent to -7*6^4 mod 23004

11701*2 is congruent to 43 mod 71

so 69660 is congruent to -(6^2-1)*6^4 mod 23004

pg(51456) and pg(92020) are primes
51456 and 92020 are congruent to 10^m mod 41

51456 is divisible by 4288
92020 is divisible by 428

4288 and 428 are numbers of the form (386*10^n-8)/9

pg(51456) is prime

51456 is divisible by 2^8 and by 67

51456 congruent to -12*2^8 mod (71*3*2^8)

51456 is congruent to -12*43 mod (71*3)

Pg(67) is prime pg(359) is prime pg(92020) is prime

-67 congruent to 4 congruent to 359 congruent to 92020 mod 71

Remarkably 92020+67=71*1297 where 1297 is a prime of the form 6^k+1

you can start from

(6^4+1) congruent to 5 mod 1292

multiplying both sides by 71

71*(6^4+1) congruent to (359-4) mod (71*323)

71*(6^4+1) +355 is congruent to -288 mod (43*107)

69660 is congruent to 17^2+359=648 mod 23004

-17^2-359 mod 23004 is -22715-22645=-45360

69660 is congruent to 9 congruent to -45360 mod 71

-45360 is even congruent to 9 mod (213^2)

7740 divides 69660

-(71^2-1)=5040 is congruent to 7740 mod (71*3)

from here we have

-1008=-10^3-2^3 is congruent to 43*6^2 mod (71*36)

it holds:

((71^2-1)+71*109+1)/71+108=288

where 71*109+1=7740 divides 69660

maybe there is some connection to the fact that 215 69660 92020 541456 are congruen to +- 108 (or 288) mod 323

probably in the ring Z426 is hidden something special about these numbers

69660 is congruent to 222 mod 426

204*24 is congruent to -6^3 mod 5112

69660 has the curious factorization (387/28)*(71^2-1)

pg(51456) is prime

noting that 359 is congruent to -67 mod 426

we have

359*2^8*3 is congruent to -51456

so 51456 is congruent to 336 mod 426 and mod 5112

336 is congruent to 1 mod 67 and 51456 is multiple of 67

in the ring Z426 the multiplicative inverse of 67 is (336+1=337)

pg(6231=67*93) is prime

6231 is congruent to -159 mod 426

curious that 337*67 is congruent to 1 mod 159

pg(331259) is prime

331259 is congruent to -13 mod (43*107)

331259 is congruent to -13^2 mod (426) and mod 777

541456 is congruent to 10 mod 426 and 10 is also the residue mod 41

my sensation is that the exponents of these primes follow some ver complex logic

331259*5 is congruent to 7 mod (426)

(2^8+1)*5 is congruent to (2^3-1) mod 426

331259*5-6 is a semi-prime congruent to 1 mod 426 and congruent to 17^2 mod 4600 (4601 divides 92020)

331259*5-6=1151*1439 and the difference is 1439-1151=288=17^2-1

pg(2131) is prime

7740 divides 69660 (7740 is congruent to 1 mod 71)

2131 is congruent to 1 mod (71*3)

7740 is congruent to 2131 mod (71*79)

so 69660 is congruent to 19179 mod (71*79)

pg(19179) is prime

curiously also pg(79) is prime

69660 is congruent to 3*28^2 mod (71*79)

i have the vague idea that these exponents are not random at all

92020 is congruent to -3333 mod (71*79)

in conclusion

69660 is congruent to 19179 which is congruent to 6^6 which is congruent to 648 which is congruent to 9 mod (71*9)

92020 is congruent to 4 mod (639)

92020 is congruent to 6^6-5 mod (213^2)

6^6 infact mod 639 is 9

92020 is congruent to 2*641 mod (213^2)

641 is a prime I think it has something special (Euler? Fermat?)

641 is congruent to 2 mod 639 anyway

72^2-21^2*10=774

69660=72^2*90-(21*30)^2

(21*30)^2=396900

pg(39699) is prime and 39699 congruent to 6^6 mod (773)

(2130^2) is congruent to 1 mod 2131 , pg(2131) is prime

69660 is congruent to (60^2-1) mod 2131

pg(2131) is prime
pg(69660) is prime
pg(2131*9) is prime

2131, 69660, 19179 are congruent to (214*3^j) mod 639, for some nonnegative j

2131 mod 639 is 214=107*2

92020 is a multiple of 107

pg(331259) is prime

331259 is congruent to -6*2^6+2 mod 23004 (6*2^6+1 is Woodal prime???)

the inverse of 2^6 mod 23004 is 10

so
331259*10 is congruent to 14 mod (23004*12^2=71*6^6)

if we work mod 639

331259*10 is congruent to -5^4 mod 639

(331259*10+625)/639=72^2+1

so 331259*10 is congruent to -5^4 mod (71*9*(72^2+1))

331259*10 is congruent to 456*10 mod ((72^2+1)*(71*3^2-1))

pg(541456) is prime

541456 is congruent to 2*6^3 mod 638
69660 is congruent to 3*6^3 mod 639

331259*10 is congruent to -5^4 mod (71*(6^6+3^2))

69660 is congruent to 3 mod 107 pg(69660) is prime

107 is the multiplicative inverse of 215 mod 23004

but 69660 is also congruent to 3 mod 651.

651=(2^2-1)*(2^3-1)*(2^5-1) so is the product of (2^p-1) taken over the first three primes 2,3,5

in other words
69660 is congruent to 3 modulo the product of the first three Mersenne primes

92020=2*6^6-(6^4-4)

2*6^6 is congruent to 6^4 mod 23004

2*12^2 is congruent to 4 mod 71

i think that something is involved related to p-adic numbers

69660 is congruent to (17^2+359) mod 71

so 69660 is congruent to (5+4=9) mod 71

92020 is congruent to 359 which is congruent to (17^2-1) mod 71

so 92020 is congruent to 4 mod 71

i think that is in some way related to the fact that 92020 is congruent to (17^2-1)=2*12^2 mod 323

69660 is congruent to (17^2+18^2+(6^2-1))=80=9 mod 71

92020 is congruent to (18^2+(6^2-1))=75=4 mod 71

69660 is congruent to 2*18^2 mod 71

92020 is congruent to (2*18^2-17^2) mod 71

so 69660 is congruent to 9 mod 71

92020 is congruent to (9+66=75=4) mod 71

pg(67) is prime, 67 is congruent to -(17^2-1) mod 71

69660 is congruent to 2*17^2+2*(6^2-1) mod 71

92020 is congruen to 17^2+2*(6^2-1) mod 71

359=17^2+2*(6^2-1)

69660 is congruent to 3*(6^3+71) mod (323*71)

92020 is congruent to (6^3+72) mod (323*71)

69660 is multiple of 18^2

69660 is congruent to 18^2=3 mod 107

maybe 36*2^k is involved...

36*2^8 for example is 9216 which is 92016=71*6^4 without the 0

pg(359) is prime and 359 is -1 mod 36

absolute value of (92020/359) is a power of 2

pg(331259) is prime. 331259 is prime
curiously (maybe it is not chance) 71*648^2/90 is near to 331259

((71/10)*6^2)*(19^2-1)+4=92020

92020*359/100+(7*6^4)/10=331259

359 is of the form 323k+36

6^6+1 is divisibile by 3589 which is 323*11+36

331259 is congruent to 4667 mod 6^6

4667=6^6-323k for some k

23005*4=92020

23005 is congruent to 5 which is congruent to -18 mod 23 (23005 is a multiple of 107 and 5)

541456 is congruent to 331259 congruent to 13 mod 23

331259=92020+239239

so mod 23:

541456 is congruent to 20+16=6^2 mod 23

or equivalently

541456 is congruent to -7*8 mod (23*6^3)

541456 is even congruent to 6^2 mod (23*107*5)

I recall that 69660 is congruent to 6^6 or 3*6^3 mod (23004)

69660 is congruent to 4^2 mod 23

using CRT 541456 is so a number of the form 36+12305s

pg(36) is prime

I could conjecture that there are infinitely many pg(k) primes with k of the form 36+12305*s

(1/18)*(331259*(2^2+1)+(2^6+1))=92020=(5/18)*(331259+13)

the curious 71*72=5112=512+4600 (4601 divides 92020)

92020/5112 is very near to 18

(331259/71)=6^6/10+7/355

(331259/6^6)=71/10+7/(2^5*3^6)

-104*215 congruent to 69660 congruent to 3 mod 107

dividing both sides by 215

-104 is congruent to 18^2 congruent to 3 mod 107

104+18^2=428 which divides 92020

-104*215 is congruent to 69660 which is congruent to 3 mod 107

the inverse mod 107 of 104 is 71

so

-215 is congruent to 69660*71 which is congruent to 213 mod 107

69660*71 is even congruent to -215 mod (107*215^2)

69660*71 is congruent to -1 mod 107

equivalent to:

-(6^3-1) congruent to -69660*6^2 congruent to 213 mod 107

213+215=428=107*4 which divides 92020

324*71 is congruent to -1 mod 107

324*71=23005 which divides 92020

(6^6-1)*215 congruent to 69660 mod (43*107*5*433)

6^6 is congruent to 325=18^2+1 mod (107*433)

(6^6-1)*(6^3-1)-(x/2+3*215^2)=107x

soluion to this equation is x=92020

curious that 331259 (pg(331259) is prime) is congruent to 44 mod (71*4665)

4665 is 6^6 (46656) with the last 6 truncated

(71/10)*6^6 is congruent to (71/10)*6^4 mod 23004

so mod 23004, 331259 is congruent to 9203 mod 23004

9203=(71/10)*6^4+1.4

consider

71*6^k+14

71*6^k+14 is divisible by 10

71*6^2+14 is divisible by 257

331259 is congruent to 257 mod (71*9)

71*6^3+14 is divisible by 1535

331259 is congruent to 1535 mod (71*9)...
and so on

331259 is congruent to 71*6/10+1.4=44 mod (71*3=213)

pg(2131) is prime as well as pg(2131*9) ...2131 is prime and congruent to 1 mod 213

I note that 71*6+14=440 which is 1 less than 21^2=441

pg(441*3=1323) is prime curious

curious that 1323 is congruent to -(13^2-1) mod 213

331259 is congruent to -13^2 mod 213

becasue (71*6^k+14)/10 is congruent to -27 mod 71

331259 is congruent to -3^3 mod 71

1323 is congruent to -(3^3-1) mod 71

for example (331259+27)/71=4666

331259*(6^3-1) is congruent to 7*43 mod (71*43*18)

1535*(6^3-1) is congruent to 7*43 mod (71*43*18)

331259 is so congruent to 1535 mod (71*43*18)

1535=71/10*6^3+1.4

23005 is congruent to 559 mod 774

dividing by 43

535 is conguent to 13 mod 18

so

535*172=92020 is congruent to 4 mod 18 which is congruent to 2236 mod 18

92020=69660+2236*10

92020 is so even congruent to 2236 mod (43*18)

-7*11*239*20-20 is congruent to 92020 mod 23005

331259 is so congruent to -539*239 which is congruent to 9203 mod 23004

and 331259 is congruent to -539*239-20 which is congruent to 9189 mod 23005

331259=92020+239*1001

(331259+539*239+20)/92020=5

43*107*k-1 i think this is the key , for k integre

331259 for example:

(43*107*6-1)*6*2-1=331259

43*107*9-1=(3*6^3-1)*2^6

331259=(3*6^3-1)*2^9-5

so for example 331259 is congruent to 7*43*107 congruent to 9203 mod 23004

43*107*10 is congruent to 1 mod (139*331)

92020 is so congruent to 2 mod (139*331)

331259 is congruent to -72 which is congruent to 71*6^6 which is congruent to 259 mod (331)

71*6^6 is even congruent to -72 mod (139*331)

by the way (331259 is 331 and 331-72=259)

71*6^6 is congruent to -215+71 mod (43*107*5)

From here

71*(6^6-1) is congruent to -215 mod (43*107*5)

From here

71*217 is congruent to -1 mod (43*107*5)

From here

-69660 is congruent to 4600*(6^6-1) mod (43*5*107)

4601 divides 92020

4601*72-13=331259

71*(6^6-1) is congruent to -215 mod 23005

(6^6-1) is divisible by 5

so

71*9331 is congruent to -43 mod 4601

so

71*129 is congruent to -43 mod 4601

71*129+43=9202

331259 is congruent to (71*129+44) mod 23004

331259 is so congruent to 44 mod 71

71*9331 is congruent to -43 mod 4601

from here

9331 is conguent to -43*4277 mod 4601

so

9331 is congruent to -18^2*43 mod 4601

from here

69660=18^2*43*5 is congruent to 9331*5 mod 4601

from here 69660 is congruent to 645=(6^3-1)*3 mod 4601

69660 is congruent to -4600*(6^6-1) mod (9332)

from here

69660 is congruent to 5*4600 mod 9332=2333*2*2

43*18^2 is congruent to 129 mod 4601

331259=(43*324-129)*3*2^3-13

331259 is congruent to -3^3 mod (71*2*2333) 9332/2=2333*2

From 71*6^6 is congruent to 6^2 mod (239*5)

I derive

92020=71*6^4+4 is congruent to 5 mod (239*5)

331259 is congruent to 244=1+3^5 mod (239*5)

3*331259 is congruent to 71^2*2^5 whcih is congruent to 3*71*6^6 mod 331

-3*331259 is congruent to -71^2*2^5 which is congruent to - 3*71*6^6 mod 331

-3*331259 is congruent to 6^3 mod 331

so
6^3 is congruent to -71^2*2^5 which is congruent to -3*71*6^6 mod 331

this leads to 1 is congruent to -3*71*6^3 mod 331

-3*331259 is congruent to 6^3 mod (331*1001)

71*6^2 is congruent to 239 mod 331

71*6^2*1001 is congruent to 257 mod 331

so

239*1001 is congruent to 239*6^4-2 which is congruent to (331259-2) whcih is congruent to 257 mod 331

so

239*1001 is congruent to 3*239*101-2 which is congruent to 331259-2 whcih is congruent to 257 mod 331

331259=92020+239*1001

so

0 is congruent to 239*3*101-2-239*1001 which is congruent to 92020-2 whcih is congruent to 257-239*1001 mod 331

so

0 is congruent to 239*(-698)-2 which is congruent to 92020-2 mod 331

or

0 is congruent to -239*6^2-2 which is congruent to 92020-2 mod 331

so 92020 is congruent to 2 mod 331

so

0 is congruent to 92*6^2-2 which is congruent to 92020-2 mod 331

so

0 is congruent to 3312-2 which is congruent to 92020-2 mod 331

331259 and 3371 are primes of the form 59+3312s

pg(331259) and pg(3371) are primes

92020 is so congruent to 3312 mod 331

so

23005 is congruent to 23*6^2 mod 331

from this the curious thing:

92020 is congruent to 92*6^2 mod 331

92020=4601*20

4601 is congruent to 298 mod 331

541456 is congruent to 298*72-1=21455 mod 331

21455 is congruent to 271 mod 331

it holds:

331259+85 is congruent to 72 which is congruent to -71*6^6-72 mod 4601

so

4673 is congruent to 72 which is congruent to -4457-72 mod 4601

4673+4457+72=9202

331259 is congruent to 9203 mod (4601*5-1=23004=71*324)

-71*6^6 is congruent to 1656*72 which is congruent to 43*107*10*72 which is congruent to -331259 mod 331

92020 is congruent to 2 mod (139*331)

71*6^6 is congruent to -46010*72 mod (139*331)

71*6^6 is congruent to -92020*36 mod (139*331)

259 is congruent to -46010*72 which is congruent to -92020*36 which is congruent to 71*6^6 which is congruent to 331259 which is congruent to 590 mod 331

92020*36+590 for example is 3313310=(331259+72)*10

92020*6^2 is congruent to -259 which is congruent to -331259 which is congruent to -71*6^6 which is congruent to 72 mod 331

so

2*6^2=72 is congruent to -259... mod 331

in the ring Z331 the multiplicative inverse of 215 is 214

so 92020=215*214*2 is 2 mod 331

in Z139 the multiplicative inverse of 215 is 214

so 92020 is also 2 mod 139

for example 331259 is congruent to -(4*9203+1) mod (139*331)

curious that 69660+22360=92020

22360 are the first five digits of the decimal expansion of 1/(5*sqrt(2))

so 92020=69660+10^n/(2*sqrt(5)) for some n

92020=2*6^6-6^4+4

(215*107*9-1)*2^4-2^7=71*6^6

(215*107*9-1) *2 is congruent to 7 mod (331*139)

so

92020 is congruent to 7*((215*107*9-1)^(-1)) which is congruent to 2 mod (139*331)

Last fiddled with by enzocreti on 2021-07-10 at 20:53 Reason: 3

 2021-07-11, 10:02 #10 enzocreti   Mar 2018 21B16 Posts Now i consider 215*107*x is congruent to 1 mod (331*139) the solution is x=2+46009*s, for some s when s=8 (215*107*(2+46009*8)-1)/(331*139)=429^2=2*92020+1 so 92020=(215*107*(2+46009*8)-1)/(331*139*2)-1/2 starting from 215*107*(2+46009*(6^6-1)) is congruent to 1 mod (331*139) i arrived to 69660 is congruent to 107*2^6*3^7 mod (331*139) form this 2*430 is congruent to (430)^(-1)*12^3 mod (331*139) Last fiddled with by enzocreti on 2021-07-12 at 14:21
2021-07-11, 17:32   #11
rudy235

Jun 2015
Vallejo, CA/.

3·383 Posts

Quote:
 Originally Posted by enzocreti pg(69660), pg(92020) and pg(541456) are probable primes with 69660, 92020 and 541456 multiple of 86 69660 in binary is 10001000000011100 92020 in binary is 10110011101110100 541456 in binary is 10000100001100010000 you can see that the number of the 1's is always a multiple of 5 a chance?

I never quite understand what you are trying to say. Can you explain to the unwashed masses the function pg? What is it? Paying guest? Picogram?

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