Near Wieferich primes

For elMath.org: Miroslav Kures
Institute of Mathematics, Brno University of Technology
Contact: kures@fme.vutbr.cz
Creation date: 2008-01-14

Near-Wieferich primes

Let us express numbers (mod p2) in a form

Z+Ap

with both Z, A reduced (mod p).


Wieferich primes are defined by

2p-1 ≡ 1 (mod p2),

i.e. 2p-1=1+0p. Thus, for Wieferich primes

2 ((p -1)/2) =1+0p (mod p2) or 2 ((p -1)/2) =p-1+(p-1)p (mod p2)

is satisfied. The second case can be written also as -1+(-1)p, because -1≡ p-1 (mod p). ”Absolute value“ |A| of A is in fact considered as

min{|A’|,|A’-p|}

(where A’ is nothing but A viewed as integer).


Now, we define near-Wieferich primes as primes having in the form Z+Ap (mod p2)

Z=±1 and |A|<=100. (128 instead 100 appears in our application.)


There are some probabilistic thinking about a number of near-Wieferich primes. According to one of these, a number of near-Wieferich primes in the range [1015,2*1015] could be cca 4. But it is a conjecture only. Up to now, two near-Wieferich primes are known in [1015,2*1015]:

1140417231387373, where Z = −1 and A = −82

1170553064286511, where Z = +1 and A = −84


Author: Miroslav Kures