mersenneforum.org - Pg primes pg(k) with k congruent to 10^n mod 41

I Pg(215) pg(51456) pg(69660) pg(92020) pg(541456) are probable primes with 215 51456 69660 92020 541456 congruent to 10^n mod 41

215 51456 69660 92020 541456 are congruent to 19*3*2^q+1 or - 1 mod 13 for some q

For example 51456 is congruent to 457=57*8+1 mod 13

51456 is congruent to (1763-307-999) mod 13

69660, 92020 and 541456 are congruent to the prime 227 mod 13
51456 is congruent to the prime 457 mod 13

227 and 457 are primes of the form 19*3*2^q plus or minus one

215 (odd) is congruent to - 227 mod 13

227 and 457 are primes of the form 10k+7

227 can be written also as 2*(58369-1)/(19*3^3-1) -1

where 58369 is a prime of the form 19*3*2^q+1

the prime 58369 can be written also as (307*10+2)*19+1

307*19 divides numbers of the form 18^(3+6k)+1

227 can be written as (3*19*2^10+2)/130-13*17-1

457 can be written as (3*19*2^10+2)/130+(307*19-1)/3^6

x=(3*19*2^10+2)/130-(215+x)/2-1

solution of this equation is x=227

19*((541456-215)/1763)-1=18

69660, 541456 and 215 are involved in this equality:

(((18*(69660/215)+1))/19)*1763=541456-215

the prime 227=1763-3*2^9 where 3*2^9 is congruent to 1 mod 307

1763=((2^2)*19+19*(3^3)-1)*3-1

the prime 457=1763+1-10^3-307

so the prime 227 is congruent to 1763-1 mod 307
the prime 457 is congruent to 1763+1-10^3 mod 307

307=((19*31-1)*3+999)/9 from which follows that

307*9 is congruent to 10^3 mod 1763

541456*10-1=307*3*5879

the prime 5879=10*((1763+1)/3-1)

(921/10)*x+(1/10)-(42^2)/3=92*x

the solution is x=5879

921=3*307=92*10+1

so the above can be written as

((92*10+1)/(10))*x+1/10-(42^2)/3=92*x

the prime 457=(42^2)/3-1 -10*(42/3-1)

541456=(((42^2/3-1)*10-1)*3*307+(307*3+1)*10-9)/10
Because 307=111+(42/3)^2
541456=(((42^2/3-1)*10-1)*3*(111+(42/3)^2)+10*(3*(111+(42/3)^2)+1)-9)/10

541456 is congruent to (42^2/3=588) mod the prime 5879=10*(42^2/3-1)-1

and (10/5879)*(541456-588) is congruent to -1 mod 307

((10/5879)*(x-588)+1)^2-307000+215-x=0

the solution to this 2-nd order equation is x=541456

Because 903 is congruent to 6 mod 13

215 69660 92020 541456 are congruent to plus minus 903 mod (13*43)

51456 is congruent to 28 mod 903

So 215 51456 69660 92020 541456 are congruent to plus or minus 7k mod (13*43) for some k

215 69660 92020 541456 are congruent to plus or minus 43*2^3=344 (mod 13*43)
51456 is congruent to (7*2^2) mod (13*43)

so 215 51456 69660 92020 541456 are congruent to plus or minus p*2^q mod (13*43) with some q and p a prime of the form 6s+1

344 is one of the residues such that 1763k+344 is a number congruent to 10^n mod 41. and 344 has the form p*2^q with p a prime of the form 6s+1

It is curious that

(215+903)/13/43=2
(92020-903)/13/43=163
(541456-903)/13/43=967

where 2,163,967 are primes belonging to the oeis sequence [URL="https://oeis.org/A162904"][COLOR=#0066cc]A162904[/COLOR][/URL]

903 is one of the residues such that 1763k+903 is congruent to 10^n mod 41

(903-344)=13*43

Finally
215, 51456, 69660, 92020, 541456 are congruent to + or minus (p^3+1) mod 559

because 28=3^3+1
and 344=7^3+1

p is a prime of the form 2^j-1

infact 28=(2^2-1)^3+1
344=(2^3-1)^3+1

I forgot the term 51

pg(51) is prime and 51 is 10 mod 41

51 is congruent to -127*2^2 mod 559 where 127 is a prime of the form 6s+1

k now i think I am ready for the new conjecture

51, 215, 51456, 69660, 92020, 541456 are comgruent to + or minus (q^3+1) mod 13 where q is a non negative integer of the form (2^j-1)

51 is infact congruent to -(0^3+1) mod 13
215 is congruent to -(7^3+1) mod 13
51456 is congruent to (1^3+1) mod 13 or (3^3+1) mod 13
69660 is congruent to (7^3+1) mod 13
92020 is congruent to (7^3+1) mod 13
541456 is congruent to (7^3+1) mod 13

note that 215, 51456, 69660, 92020, 541456 are also congruent to plus minus (q^3+1) mod 43 with q odd
in the case of 51 q is even and 51 is congruent to 2^3 mod 43

so the conjecture is:

when pg(k) is probable prime and k is congruent to 10^m mod 41 for some m, then pg(k) is also congruent to plus or minus (2^q-1)^3+1 (mod 13) for some nonnegative q!

Restatement of the conjecture:

Conjecture : terms of the sequence that are congruent to 10^m mod 41 for some nonnegative integer m and congruent to + or - q^3 (mod 43) for some q positive integer, are congruent to + or - j^3 (mod 13), for j some positive integer. Terms of the sequence that are congruent to 10^m (mod 41) for some nonnegative integer m and congruent to + or - (q^3+1) (mod 43) for q some positive integer, are congruent to + or - (j^3+1) (mod 13) for j some positive integer. This is the case for terms: 51, 215, 51456, 69660, 92020, 541456

terms of the sequence that are congruent to 10^m nod 41 and that are not congruent to (q^3+1) (mod 43) for every non negative q, are congruent to j^3 (mod 13) for j some positive q...terms of the sequence congruent to 10^m mod 41 and congruent to (q^3+1) mod 43, are also congruent to (j^3+1) mod 13

51 is not congruent to (q^3+1) mod 41 for every q

there is no solution to (51 is congruent to + or - (x^3+1) mod 43)
so 51 is congruent to - (1^3) mod 13

in the cases 215 51456 69660 92020 541456 there is a solution
215 is congruent to -(7^3+1) mod 43 and so 215 is also congruent to -(7^3+1) mod 13

51 215 51456 69660 92020 541456 are congruent to + or - (2^j-1)^3+1 mod 13 for some nonnegative j
51+(2^0-1)+1 is congruent to 3^2 mod 43
215+(7^3+1) is congruent to 0^2 mod 43
...
so if P denotes the generic term (P=51, 215, 51456, 69660, 92020, 541456), then

P + or - (2^j-1)^3+1 is congruent to a square mod 43

215 69660 92020 541456 which are multiple of 43 are always congruent to + or - (7^3+1) mod 559

A curio:
215 92020 541456 are multiple of 43 but not of 3

(215+344) /559+1 =2
(92020-344)/559-1=163
(541456-344)/559-1=967

Where 2 163 and 967 are primes that are 2 lesser than a tetrahedral number... 2=4-2
163=165-2
967=969-2

51456 is congruent to 28 mod 559

But 28=(1763-559)/43

51 is congruent to - 508 mod 559
508=344+41*4

So 215 is congruent to - 344 mod 559
69660, 92020 and 541456 to 344 mod 559
51 to - (344+41*4) mod 559
51456 to (41-13) mod 559

OK now something

51 is congruent to - 508 mod 559
215 is congruent to - 344 mod 559
69660 is congruent to 344 mod 559
92020 is congruent to 344 mod 559
541456-2 41446 is congruent to 344 mod 559

344 and 508 are congruent to 16 mod 41

Maybe better

The residues mod 559 are: 28, 508, 344

51 is congruent to - 508 mod 559
215 is congruent to - 344 mod 559
51456 is congruent to 28 mod 559
69660 is congruent to 344 mod 559
92020 is congruent to 344 mod 559
541456 is congruent to 344 mod 559

The residues mod 559 are 28, 508, 344

28, 508 and 344 are congruent to 2^j mod 12 for some j

541456 is congruent to - 215 mod 559

215+344=559

215, 344, 903 leave a residue 6 mod 13

The multiples of 43
215, 69660, 541456, 92020 are congruent to plus or minus (or 215 or 344 or 903) mod 559

215, 69660, 541456, 92020 are numbers of the firm 1763k+(903 or 215 or 344)

51 is congruent to 12 mod 13
51 is congruent to - 508 mod 559
508 is congruent to - 12 mod 13

51456 is congruent to 2 mod 13
51456 is congruent to 28 mod 559
28 is congruent to 2 mod 13

69660, 92020, 215, 541456 are congruent to plus or minus 6 mod 13
And they are congruent to 344 mod 559
344 is congruent to 6 mod 13

51 is - 1 mod 13...the absolute value of - 1 is 1
51+(13x+1) is congruent to 0 mod 43 for x=39
And 51+(13*39+1) is congruent to 0 mod 559

51456 is congruent to 2 mod 13
51456-(13x+2) is congruent to 0 mod 43 for x=2
And 51456-(13*2+2) is congruent to 0 mod 559

215, 69660, 92020, 541456 are congruent to plus or minus 43*8 mod 559
51 is congruent to minus 43*8+41*4=508 mod 559

51456 is congruent to 28 mod 559

But 28 =(1763-559)/43

559 is the residue mod 28 and mod 43 of 1763

51 is congruent to - 508 mod 559
508=43*10-11*2^5

28=43*10-13*2^6

51456 is also congruent to 888 mod 43*28

1763-43x is congruent to 0 mod 559 has the solution x=13n+2

For n=1 you have 28
Which is the residue mod 559 of 51456

51456 is a number of the form 1763k+329
329 mod 43 is 28
51456 is 28 mod 559

(41*29+7)*43+28=51456

51456=(1763-9*63)*43+28
541456=(1763-3*63)*43*2^3

51 is congruent to -508 mod 559

508=(1763-9*63)-43*2^4
215, 69660, 92020, 541456 are congruent to 344 mod 559. but 344=((1763-9*63)-508)/2

51456 is a multiple of 67 it has the form 67*41*18+2010
2010 is 2^5 mod 43
51 is multiple of 17 and it is of the form 17*41k+51.
51 is 2^3 mod 43

51456 and 51 are not multiple of 43 and congruent to 10^m mod 41

51456 (even) is congruent to 111 mod 163
51 (odd) is congruent to -(111+1) mod 163

so i could conjectur that if pg(k) is prime and k is 10^n mod 41 and not multiple of 43, then

k is congruent to 111 mod 163 if k si even
k is congruent to -(112) mod 163 if k is odd

541456 mod 163 =133

133 is the same residue of 1763 mod 163
41*3^4*163-307*1763=2*41

541456 is congruent to 1111 mod 163

so 51456=63*163*5+111
541456=663*163*5+1111

Exists this beautiful relation between 51456 and 541456

541456=700^2+51456=10^3*(163*3+1)+51456

51 is 51 mod 163
541456 is 133 mod 163

51 and 133 are numbers of the form 41s+10

(541456-133)/41/163=81

51, 215, 541456 are numbers of the form 41s+10

51 is 51 mod 163. 51=41+10
215 is 52 mod 163 52=41+11
541456 is 133 mod 163 133=41*3+10

51 is a quadratic residue mod 163 (41^2-51 is 163*10)

215, 69660, 92020, 541456 are congruent to + or - 344 mod (559)

559=41^2-51*22

51 is congruent to - 508 (mod 559)

508=41^2-51*23

541456 and 51456 are congruent to 6 mod 49
49=41^2-51*32

49=(163*3+1)/10

so multiples of 43: 215, 69660, 92020, 541456 are congruent to + or - 344 (mod 41^2-51*22) where 51 is a quadratic residue mod 163

41^2-51*22-215=344

10 is a quadratic residue mod 163, infact 559^2-10 is a multiple of 163

51, 215, 51456, 69660, 92020, 541456 seem to be congruent to some s+163n (mod 163)

the value of s seems to be not random

values of s are=51,52,59,88,111,133

with the exception of 88

51,52,59,88,111,133 are numbers of the form 43n+k^g with n,k,g some integers (g>1)

51=43+2^3
52=43+3^2
59=43+4^2
111=43*2+5^2
133=43*3+2^2

so with the exeption of s=88, the other values are of the form 43n+ a power

I note however that in the case of 88
88+163=251 which is of the form 43s+36 (36 is a power)

so terms that are congruent to 10^m mod 41

51,215,51456,69660,92020,541456 are conguent to (43*j+p^q) (mod 163) for some positive j and p and q>1

if I call r the residue mod (163) so found

r=51,52,59,88,111,133

so they are numbers of the form:
or 43s+power
or 41s+power

51,52,59,111,133 are of the form 43s+power

88=41*2+2^2 is of the form 41s+power

so the conjecture is that terms k such that k is congruent to 10^m mod 41 that is 51,215,...541456

are congruent either to (41s+q) (mod 163) or to (43s+q) (mod 163) where q is a power

there is a mistake

88 is not 41*s+power

51, 215, 69660, 51456 and 541456 leave a residue 10^m mod 41
92020 leave a residue 16 mod 41

51 is congruent to 51=43+8 (mod 163)
215 is congruent to 52=43+9 (mod 163)
69660 is congruent to 59=43+16 (mod 163)
51456 is congruent to 111=43*2+25 (mod 163)
541456 is congruent to 133=43*3+4 (mod 163)

whereas 92020 leave a residue 16 (mod 41) (and also a residue 10^3 mod 41)
92020 mod 163 is equal to 88, which is not of the form 43*s+power...

92020 is not of the form 41s+10^m with m<3, but of the form 41s+16

92020 is congruent to a square=16 mod 41

so the conjecture now is that
terms congruent to 10^m mod 41, that are not congruent to 2^j (with 2^j a square) mod 41, are congruent to 43s+p (mod 163) where p is a power...

92020 is congruent to Fermat 2^(2^2)=16 mod 41

and 92020 is 16 mod Fermat prime 17

so 92020 is congruent to (p-1) mod p

where p is a Fermat prime

I think it is quite surprising that terms of the sequence, that reduced mod 41 are 10^m, are congruent to 43s+p (mod 163) where p is a power>1

This is the vector of k such that pg(k) is prime

[2, 3, 4, 7, 8, 12, 19, 22, 36, 46, 51, 67, 79, 215, 359, 394, 451, 1323, 2131, 3336, 3371, 6231, 19179, 39699, 51456, 56238, 69660, 75894, 79798, 92020, 174968, 176006, 181015, 285019, 331259, 360787, 366770,...541456]

I think that for k>43

the k's which are congruent to a number of the form 43s+p (with p a power>1 and s >0) modulo 163 are 51,215,3371,51456,69660,541456.

3371 is prime and reduced mod 41 is 9=10-1
3371 mod 163 is 111=43*2+25

51,215,51456,69660,541456 reduced mod 41 are 10^n...

so k's reduced mod 41 that are 10^n or 10^n-1, are congruent to a number of the form 43s+p (mod 163)

Let me point out this astonishing fact

The multiples of 43 are 215, 69660, 92020 and 541456

215, 69660, 92020 and 541456 are congruent to

+ or - (7^3+1^3) mod (6^3+7^3)

51 is congruent to - 508 mod (7^3+6^3)

But 508=127*4=(7^3-6^3)*4 where 127 is a Mersenne prime

I note that 1^3, 6^3 and 7^3 are cubes congruent to + or -1 mod 43

Because 344=6^3+2^7

215 69660 92020 541456 are congruent to + or - (2^7+6^3) mod (6^3+7^3)

6^3 and 7^3 are two cubes congruent to 89 mod 127
So 215 69660 92020 541456 are congruent to something like + or - (127+217) mod (127*3+89*2)

89 is a quadratic residue mod 559=6^3+7^3

215, 69660, 92020, 541456 are congruent to + or - 11303=89*127 mod 13

Astoounding fact

(541456-11303) /169-1=56^2

89 and 127 are Mersenne exponents

541456 mod (127*89) is 10^4+6^3-1

541456-10215=47*127*89
A122094 Prime divisors of Mersenne numbers. Primes p such that the multiplicative order of 2 modulo p is prime. +30
12
3, 7, 23, 31, 47, 89, 127, 167, 223, 233, 263, 359, 383, 431, 439,

6^4+7+10000=127*89=10007+6^4 where 10007 is prime of the form 13s+10

Now

(541456-11303)/13^2
(92020-11303)/13
(69660-11303)/13
(215+11303)/13
Leave a power of 2 as a residue mod 13

Let me point out this relevant fact

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

But the astounding fact is that
344=(67^2-17)/13
559=(67^2-17)/8

In other words lcm(344,559)=67^2-17

The greatest factors of

541456-(67^2-17)
215+(67^2-17)
69660-(67^2-17)
92020-(67^2-17) are primes with primitive root 10

215, 69660, 92020, 541456 are congruent to + or - (10^4+12*127) mod 559

where (10^4+12*127) is congruent to (7^3+1) mod 559

11524=10^4+12*127 is congruent to (2^7+1) mod (6^3-1)

215, 69660, 92020, 541456 are simply congruent to + or - 51*215 (mod 559)

215, 69660, 92020, 541456 are congruent to + or - 11524 mod 559

In Oeis sequence [URL="https://oeis.org/A282620"][COLOR=#0066cc]A282620[/COLOR][/URL] and [URL="https://oeis.org/A050549"][COLOR=#0066cc]A050549[/COLOR][/URL] 11524 is present.

11524 is a number such that 47*2^11524-1 is prime
11524 is a number such that divides 5^11524+3
11524 is a number such that (67*10^11524 + 17)/3 is prime. Numbers naer repdigits of the form 223333...39

215+2^7 is congruent to 0 mod 7^3
69660-2^7 is congruent to 14^2 mod (6^3)
92020-2^7 is congruent to 6^3 mod (1763)
541456-2^7 is congruent to 2^5 mod (6^3)

ASTOUNDING FACT:

92020 and 541456 mod (6^4) are powers of 2!!!

in particular 541456 mod (6^4) is 2^10!!!

yes now it's clear ...

51, 215, 51456, 69660, 92020, 541456 reduced mod (6^k) for some k are numbers of the form

(2^z*3^y) for some nonnegative z,y

-51 mod 6 is 2^0*3^1
-215 mod 6 is (2^0*3^0)
51456 mod 36 is (2^2*3)
69660 mod 216 is (2^3*3^3)
92020 mod 216 is (2^2*3^0)
541456 mod 1296 is(2^10*3^0)

69660-2^7 is congruent to 775 mod 1763
92020-2^7 is congruent to 216 mod 1763

775 and 216 are numbers of the form 559k+216

541456+2^7 and 215+2^7 are congruent to 7^3 (mod 1763)

92020+2^7 is congruent to 472 mod 1763
69660+2^7 is congruent to 1031 mod 1763
215-2^7 is congruent to 87 mod 1763
541456-2^7 is congruent to 87 mod 1763

Numbers of the form 1763*s+903 and 1763*s+344 are 69660 and 92020

69660-2^7 is congruent to (6^3+559) mod 1763
92020-2^7 is congruent to (6^3) mod 1763

Numbers of the form 1763s+6^3-1 are: 215 and 541456

215 and 541456 + 2^7 are congruent to 7^3 mod 1763

int(69660/1763) andi int( 92020/1763) are multiple of 13 where int is the integer part

215, 69660,92020, 541456 are numbers of the form 1763s+r
with r=215,344,903

215=6^3-1
344=6^3+128
903=6^3+128+559

-1, 128 and (128+559) are congruent to -1 mod 43

215, 69660, 92020 and 541446 are cong plus or minus 344 mod 559

344=2*43^2-559*6
2*43^2-559*6-129=215

So

215-2^7 and 541456-2^7 are congruent to 87 mod 1763
87=2*43*43-559*6-128*2-1

92020-2^7 is congruent to 2*43*43-559*6-128 mod 1763

69660-2^7 is congruent to 2*43*43-559*5-128 mod 1763

So 215 and 541456 are congruent to 2*43*43-559*6-129 mod 1763
92020 is congruent to 2*43*43-559*6 mod 1763
69660 is congruent to 2*43*43-559*5-128 mod 1763

The residues mod 1763 of 215, 69660, 92020 and 541456 are 215, 344, 903...
But 215, 344, 903=absolute value of (2*43*43-559k)
For k=5,6,7

This imply that 215, 69660, 92020 and 541456 are congruent to + or - 2*43^2 mod 559

215, 69660, 92020 and 541456 are congruent to + or - (6^4+7^4+1) mod (6^3+7^3)

51 and 51456 which are not multiple of 43 but congruent to 10^m mod 41 are congruent to - (6^4+7^4+1) mod 23 and to - (6^4+7^4+2) mod 15

215, 69660, 92020, 541456 are congruent to abs(344 + or - 559k) mod 1763

where abs is the absolute value

51456 is a number of the form 2747k+2010, because it is congruent 10^m mod 41

2010 mod 559 is 333

51456 is a number of the form 2747k+2010=2747k+1677+333

1677=3*13*43
1677 mod 41=37 which is the residue mod 41 of 10^4
333 is a multiple of 37

so 51456=41+37w+2747k for some w,k

215, 69660, 92020, 541456 are numbers of the form 1763x+344 + or - (16y-1), where 16 is the residue of 10^3 mod 41

infact 215, 69660, 92020, 541456 are numbers of the form 1763s+ (344) or 1763s+(344+559) or 1763s-(344-559)
559 is congruent to -1 mod 16

so 559 i scongruent to -1 mod (16) and mod (10)

559 is a number of the form 79+80k

51456(multiple of 67 and congruent to 10^m mod 41)=67*41*k+1677+333

1677 mod 41 is 37, which is one of the residues of 10^n mod 41
so 51456=67*41*k+1677+37*s

541456=1763*k+|344-559|

344 mod 41 is 16, which is one of the residues of 10^n mod 41

so
541456=1763*k+|344-(16*y-1)|

anologous for 215, 69660, 92020

multiples of 43, 215 69660 92020 and 541456 are congruent to + or -

41*2^n+16 mod (16m-1) for some m,n

41*2^n+16 mod 559

for n=-215, +69660,+92020,+541456

the result is always the same

n=84s+3

Let me point out this fact

Look at 51456 and 541456

51456 is multiple of 67
541456 is multiple of 43

51456-2747-1677 is a multiple of a very curious prime 5879
541456*10-1=5414559 (559=13*43) is a multiple of 5879.
5879*3=17637 is the concatenation in base 10 of 1763 and 7

51456 is congruent to 4424 mod 5879
4424=4423+1 where 4423 is a prime

5879-4423=1456!

4423 isa special prime because
2^4423-1 is prime so it is a Mersenne prime!

(541456-(1763-344)) mod 5879 is 4*243
92020-1763-344) mod 5879 is 1728=64*27

541456 is congruent to 3*14^2 mod 5879=30n^2-1

51456 is congruent to 4424 mod 5879
541456 is congruent to (4424-3836) mod 5879
92020 is congruent to (4424-589) mod 5879

69660 is congruent to (4424-567) mod 5879

So 51456, 69660, 92020 541456 are congruent to

(4424-g) mod 5879 where g is a number congruent either to 0 or 1 mod 7

3836, 4424, 567, 587 are congruent either to 16 or 17 mod 19

541456 is congruent to (4423+1) mod 5879

92020 is congruent to (4423-3*14^2) mod 5879
541456 is congruent to 3*14^2 mod 5879

51456 (which is not multiple of 43) is congruent to 3*n^2-1=587 mod 559
-51(not multiple of 43) is congruent 508 mod 559
508=3*13^2+1

So the not multiple of 43 when even (51456)
Are congruent to 3*n^2-1 mod 559
When odd (51) are congruent to -3*n^2-1 Mod 559

So I have to find another exponent congruent to 10^n mod 41 to see if the pattern holds

so...exponents congruent to 10^m mod 41:

either are congruent to + or - (7^3+1) mod 559 if multiple of 43

or

are congruent to + or - 3*n^2-1 mod 559 if NOT multiple of 43

508 and 587 have also this property:

508^12 is congruent to 1 mod
[URL="http://factordb.com/index.php?id=27203085"][COLOR=#002099]27203085[/COLOR][/URL]
587^12 is congruent to 1 mod
[URL="http://factordb.com/index.php?id=27203085"][COLOR=#002099]27203085[/COLOR][/URL]

Maybe this is better:

51456 is congruent to +(28=3*3^2+1) mod 559
51 is congruent to -508=-3*13^2-1 mod 559

so the non-multiple of 43 are congruent to

+ or - (3*n^2+1) mod 559 for some n

in the cases found n=3,13 which are primes

it is curious that

51456 is also congruent to (2^5-4) mod 559
51 is congruent to - (2^9-4) mod 559

so it could be that non multiples of 43 are congruent to + or - (2^n-4) mod 559 for some n

51 is also congruent to -(344+3^5-51-28) mod 559
51456 is also congruent to (344+3^5) mod 559

51, 215, 51456, 69660, 92020, 541456 are congruent to + or - N mod 559

where N is a number such that 2^(phi(N)) mod N is a power greater than 1

so for example 51 is congruent to -508 mod 559

2^(phi(508)) mod 508 is a power greater than 1

the same for 344 and 28

phi is the Euler totient function

Let me point out this fact

Even k congruent to 10^m mod 41 that is

51456, 69660, 92020, 541456 are of the form 5879*s+r
Where r is a multiple of 7 or 13 (in the case of 92020) and s is an integer

In the case of 92020=15*5879+3835

3835 is multiple of 13 and congruent to - 1 mod 7

so 51456, 69660, 92020, 541456 are of the form 5879s+r where r is an integer either congruent to 0 or - 1 mod 7
5879 mod 3835 is a multiple of 7

Numbers 51456, 69660, 92020, 541456 are numbers of the form 5879s+r

where r is either a number of the form (17*k+10) or (17*k+4)

in particular the even multiple of 43 that is 69660,92020,541456 are numbers of the form 5879s+(17k+10)

in particular multiples of 43,

215, 69660, 92020, 541456 are numbers of the form (5879s+17k+10) if the number is even (cases 69660, 92020, 541456)
in the case 215 (odd) is a number of the form (5879s+17k+11) for s=0

so 215=5879*0+17*12+11
69660=5879*11+17*293+10
92020=5879*15+17*15^2+10
541456=5879*92+2*17^2+10

0,11,15,92 are congruent to a square mod 7

0 is congruent to 0^2 mod 7
11 is congruent to 2^2 mod 7
15 and 92 are congruent to 1^2 mod 7

(12-0) is a multiple of 6
(293-11) is a multiple of 6
(15^2-15) is a multiple of 6 and a primorial=210
(92-2) is a multiple of 6

so multiple of 43

215, 69660, 92020, 541456

are either of the form (5879*r+m*17^q+10) (in the case of even numbers) or the form (5879*r+m*17^q+11) (in the case of odd numbers)

where |m-r| is a multiple of 6

r is ome nonnegative integer q>0,m>0

5879 is a prime of the form 30n^2-1

because 5879=559*10+17^2

215 is of the form

((5590+17^2)*r+m*17^q+11)
69660, 92020, 541456 of the form
((5590+17^2)*r+m*17^q+10)

now look at multiples of 43 which are even

(69660-5590*11-2*17^2-10) is a number divisible by 34 and 223 (r=11)
(541456-5590*92-2*17^2-10) is a number divisible by 34 and 23 (r=92)
(92020-5590*15-2*17^2-10) is a number divisible by 34 and 223 (r=15)

223 and 23 are primes ending with digits 23

so 69660, 92020, 541456 (even multiple of 43) have the form

p*34^k +5590r+588

where p is a prime ending with 23
and k is some integer

23 and 223 are primes of the form (2*10^3+7)/9

69660-5590*11-588=34*223
92020-5590*15-588=34*223
541456-5590*92-588=34*(223+559)=23*34^2

69660=5879*11+17*293+10
92020=5879*15+17*15^2+10
541456=5879*92+2*17^2+10

r=11,15,92

I note also that multiple of 43

215, 69660, 92020, 541456 are congruent to + or - (1 or 11) (mod 17)

69660, 92020 and 541456 are of the form

p*34^k+5879*r-(r-2)*17^2+10

where p is 23 or 223.

k is and integer and r=11,15,92

223=34*23-559

215, 69660, 92020 and 541456 are congruent to + or - (344) mod (34*23-223)

215, 69660, 92020, 541456 are congruent to + or - 344 mod ((34*2*10^2+33*7-2*10^3)

I note that 344 is a quadratic residue mod 223

34 is 11 mod 23
344 is 11^2 mod 223

559=((31*10-4)*(2*10^2+7)-9*(2*10^3+7)) /81

(44*10^3-8*10^2+217*10-91)/81=559

92020, 69660, 541456 are quadratic residue either mod 223 or mod 23^2

-215 is a cubic residue mod 223

so it seems alternating mod 223 and mod 23

-215 (odd) is a cubic residue mod 223
69660(even) is a quadratic residue mod 23
92020(even) is a quadratic residue mod 223
541456(even) is a quadratic residue mod 23

-215 is 8 mod 223
69660 is 19^2 mod 23^2
92020 is 12^2 mod 223
541456 is 17^2 mod 23^2
I note that

92020 and -215 which are not congruent to a^2 mod 23^2, are congruent to 20 and 15 mod 23

so 92020 is congruent to 20 (last two digits of 92020) mod 23
-215 is congruent to 15 (last two digits of 215) mod 23

in other words 541456 and 69660 are quadratic residues mod 529

92020 and 215 are non quadratic resiudes mod 529

92020 is congruent to 20 mod 23
-215 is congruent to 15 mod 23 (last two digit of 215)

69660, 92020 and 541456 have the form 23*34^2+559x+29

[215, 69660, 92020 541456] =N are congruent to + or - m mod (41*43*13)

m is either a number of the form 215s+129 for some s if N is of the form 1763k+r (with r=344,215) or a number of the form 215s+43 if N is a number of the form 1763k+903

215, 92020, 541456 are numbers of the form 1763k+r with r not multiple of 3 (r can be or 215 or 344)

numbers of this form are congruent either to 344 or 14319 mod 22919. I note that both 344 and 14319 are congruent to 2^n mod 343 for some nonnegative n

Aftter long efforts

I NOTE that 215 69660 92020 and 541456 are either of the form 215*(13s-1) or 344*(13s+1)

Noting that numbers congruent to + or - 344 mod 559 congruent to 10^m mod 41 can have only five different forms

215 69660 92020 541456 are either of the form (+ or -344k)+41*559s for some k and s or (+ or - (215k+344)+41*559s) for some k and s

so if I am not wrong 215, 69660, 92020, 541456 are either of the form (344s) +22919k for some s and k or of the form (344s plus 22919k+215)

so

215= 344*0+22919*0+215

69660=+(344*2+215)+22919k...

lcm(215,344,559)=22360

69660, 92020, 541456 are numbers of the form 344+2236x

x is congruent to 0 mod 344, x is congruent to 344 mod 559, x is congruent to - 559 mod 215, x is congruent to 10 mod 41

solution is x=541456

69660, 92020, 541456 are either congruent to 0 or 172 mod 344. 172 is gcd(344,2236)

I note that 92020 and 541456 which are multiple of 86 and not multiple of 3 are congruent to 5590 mod (3*67*43)

69660, 92020, 541456 (multiple of 86) are congruent to 10^s*(516+43*s) mod (3*67*43)

92020=10*(8643+559)
69660=10*(8643-559*3)

541456 mod 8643 is 559*10

8643=(2*10^2+1)*43=3*67*43

So 92020 and 541456 which are not multiple of 3 are congruent to 559*10 mod 8643

whereas 69660 which is multiple of 3 is congruent to - (559*10*3) mod 8643

So 215, 69660, 92020 which are multiple of 215 are numbers of the form 2580 + or - 2795n, whereas

541456 which is not multiple of 215 but of 43 it is a number of the form (2580-559)+2795n

215 (odd) is congruent to - 1462 mod 1677
92020 and 541456 (even and not multiple of 3) are congruent to 1462 mod 1677
69660 (even and multiple of 3) is congruent to (1462-559) mod 1677

So 215, 69660, 92020 and 541456 are congruent to q*(1462-559k) mod 1677

where q is 1 if the number (69660 for example) is even and q=-1 if the number is odd and k=1 if the number is a multiple of 3 and k=0 if the number is not a multiple of 3 (92020 is an example)

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

215 is congruent to 215 mod 301
69660 is congruent to 129 mod 301
92020 is congruent to 215 mod 301
541456 is congruent to 129*2 mod 301

So if the number is congruent to 0 mod 215 then the number is congruent either to 129 or 215 mod 301
If the number is not multiple of 215 (541456 is the case) then 541456 is congruent to 129*2 mod 301

69660, 92020 and 541456 are even

516 divides 69660

69660 is congruent to 516 mod (67*43*3)

516 does not divide 92020 and 541456

92020 and 541456 are congruent to 163 mod (67*3)

69660 (multiple of 3) has the form 516s
92020 and 541456 (not multiple of 3) have the form 516s+172

If i am not wrong

215, 92020, 541456 (not multiple of 3) are congruent to + or - (559*3-215) mod (3*559)

69660 (multiple of 3) is congruent to (559*2-215) mod (3*559)

69660 (multiple of 3) is congruent to (1763-215)/3 mod (67*43)
92020 and 541456 (not multiple of 3) are congruent to (1763-215*4)*3 mod (43*67)

69660 is congruent to (1677-(344+43)*3) mod (67*43)
92020 and 541456 are congruent to (1677+344*3) mod (67*43)

215, 344, 903=559*2-215, 1462=559*3-215 are all numbers of the form (1763-(7+13s)*43) for some s.

Consider the congruence:

y is congruent to 1763-43*(13x+7) mod 1677 if y is even

-y is congruent to 1763-43*(13x+7) mod 1677 if y is odd

Using Wolphram I see that 215, 69660,92020,541456 are solutions y to the congruence

So for example if y=215 there is a solution to

-215=1763-43*(13x+7) mod 1677

If y=69660 (even) there is a solution to

69660=1763-43*(7+13x) mod 1677

344 is the least positive integer y satisfying the congruence

Maybe it is chance

-215 is congruent to 43*2 mod 301
69660 is congruent to 43*3 mod 301
92020 is congruent to 43*5 mod 301
541456 is congruent to 43*6 mod 301

215 92020 541456 are congruent to plus or minus 6^4+7^4+1-559*4 mod (559*3)
69660 is congruent to 6^4+7^4+1-559*5 mod (559*3)

92020, 69660 are congruent to 172 mod 344

172=6^4+7^4+1-1763*2

541456 is 0 mod 344=2*(6^4+7^4+1-1763*2)

92020 and 69660 are congruent to (215*9-1763) mod 344

215, 92020, 541456 (not multiple of 3) are congruent to plus or minus (1763-301) mod 1677
69660 (multiple of 3) is congruent to (1763-301-559) mod 1677