[seqfan] Re: Wieferich primes and A049096

Max Alekseyev maxale at gmail.com
Fri Nov 20 22:59:32 CET 2015


Hi Robert,
PrimeGrid has already reached 4.974*10^17 as the lower bound for the next
Wieferich prime.
Max



On Fri, Nov 20, 2015 at 4:04 PM, <israel at math.ubc.ca> wrote:

> A049096 is: Numbers n such that 2^n + 1 is divisible by a square > 1.
>
> I was able to prove that this is equivalent to:
>
> Numbers n such that gcd(n, 2^n + 1) > 1 or n = k m where k is odd and 2 m
> is the order of 2 modulo a Wieferich prime.
>
> There are only two known Wieferich primes (1093 and 3511), of which only
> 1093 produces members of this sequence (182 k for odd k), since the order
> of 2 mod 3511 is odd. The next Wieferich prime, if any, is greater than
> 6.7*10^15.
>
> Now it's not easy to check directly whether 2^n + 1 is squarefree, because
> this requires factoring the large number 2^n + 1. On the other hand, the
> condition gcd(n, 2^n + 1) > 1 is easy to check, even if n is in the
> millions, because we can compute 2^n + 1 mod n without computing 2^n. But
> what about the other case? Is there a nice lower bound on those m such that
> 2m is the order of 2 modulo a Wieferich prime > 1093? The fact that 2^m+1
>
>> = p^2 where p > 6.7*10^15 only gives you m > 104.
>>
>
> Cheers,
> Robert
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>



More information about the SeqFan mailing list