Near Wieferich primes

Let us express numbers (mod p^{2}) in a form
Z+Ap
with both Z, A reduced (mod p).
Wieferich primes are defined by
2^{p}^{1} ≡ 1 (mod p^{2}),
i.e. 2^{p}^{1}=1+0p. Thus, for Wieferich primes
2^{ ((}^{p}^{ 1)/2)} =1+0p (mod p^{2}) or 2^{ ((}^{p}^{ 1)/2)} =p1+(p1)p (mod p^{2})
is satisfied. The second case can be written also as 1+(1)p, because 1≡ p1 (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 nearWieferich primes as primes having in the form Z+Ap (mod p^{2})
Z=±1 and A<=100. (128 instead 100 appears in our application.)
There are some probabilistic thinking about a number of nearWieferich primes. According to one of these, a number of nearWieferich primes in the range [10^{15},2*10^{15}] could be cca 4. But it is a conjecture only. Up to now, two nearWieferich primes are known in [10^{15},2*10^{15}]:
1140417231387373, where Z = −1 and A = −82
1170553064286511, where Z = +1 and A = −84
Author: Miroslav Kures