Primes of the form 555... 915824341
 

555915824341 is prime


No other prime found of the form 555...5915824341.

Curiously the Wieferich prime 1093 divides

55555555555555555555555555555555915824341

Certainly a low-weight sequence, but 5*(10^156-1)/9+360268786 is prime.

...
 

Finitely many?

555915824341 is a quadratic residue mod 1093

I checked the number of leading 5's out to 2000. Four of them gave (pseudo)primes.

[code]? k=915824341;for(i=1,2000,k*=10;k-=3242419069;if(ispseudoprime(k),print(i)))
3
147
179
1643[/code]

I note that if there are

1, 4, 7, ... 3*k + 1 leading 5's the number is divisible by 3

2, 8, 14, ... 6*k + 2 leading 5's the number is divisible by 13

so half the numbers are divisible either by 3 or 13.

...
 

yes I noted that too
I think that there is an explanation for that

Proof
 

can be given a proof that when the number is divisible by 557, then it is also divisible by 19 and by 13?




5*(10^59-1)/9+360268786 for example is divisible by 557 and by 19 and 13




and a proof that when the number is divisible by 1093 is divisible also by 19 and by 13?

primes of the form 8n+5
 

maybe when the numer has a small factor as 557 or 1093 (primes of the form 8n+5), then it is also divisible by 13 and 19?






so for example
5*(10^23-1)/9+360268786 is divisible by 13, 19 and 29 (prime of the form 8n+5)

[QUOTE=enzocreti;537829]can be given a proof that when the number is divisible by 557, then it is also divisible by 19 and by 13?[/quote]
Counterexample: 5*(10^337-1)/9+360268786
[quote]and a proof that when the number is divisible by 1093 is divisible also by 19 and by 13?[/quote]
Counterexample: 5*(10^314-1)/9+360268786

:sleep: