Results for Wieferich Primes

Abstract

Let v ≥ 2 be a fixed integer, and let x ≥ 1 and z ≥ x be large numbers. The exact asymptotic formula for the number of Wieferich primes p, defined by v p−1 ≡ 1 mod p 2 , in the short interval [x, x + z] is proposed in this note. The search conducted on the last 100 years have produced two primes p < x = 10 15 such that 2 p−1 ≡ 1 mod p 2 . The probabilistic and theoretical information within predicts the existence of another base v = 2 prime on the interval [10 15 , 10 40 ]. Furthermore, a result for the upper bound on the number of Wieferich primes is used to demonstrate that the subset of nonWieferich primes has density 1.