[seqfan] Wieferich primes and A049096

israel at math.ubc.ca israel at math.ubc.ca
Fri Nov 20 22:04:54 CET 2015


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



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