Thread: 69660 and 92020
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Old 2020-03-30, 23:06   #8
enzocreti
 
Mar 2018

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Default ... 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
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