For odd integer $n$ define the function $$ J(n)=(2^{\varphi(n)}-1) \bmod{n^2}$$
$J(n)$ is integer in $[0,n^2-1]$ and it is divisible by $n$.
Integer $n$ is Wieferich number iff $J(n)=0$ and if $n$ is prime it is Wieferich prime.
If $J(n)$ is odd, it is not zero and $n$ is non-Wieferich number.
We are interested in the parity of $J(n)$.
Experimentally for random $p$, $J(p)$ is odd with probability about $\frac12$.
Let $r > 1$ be positive odd integer.
Conjecture 1 For all $r$, $J(\frac{2^r+1}{3})$ is odd.
Conjecture 2 For all $r$, for at least one prime factor $q$ of $\frac{2^r+1}{3}$ we have $J(q)$ is odd. Proof of this conjecture will give infinitely many non-Wieferich primes.
Conjecture 3 For all integers $m > 2$, $J(2^m-1)$ is even.
These conjecture hold for $m,r$ up to $210$ and then factorization becomes slow.
Opportunistic open problem Is there infinite sequence of primes $a(n)$ such that $J(a(n))$ is odd?
