Integer $n$ is Wieferich number if $2^{\phi(n)}-1 \equiv 0 \pmod {n^2}$.

Wieferich prime is Wieferich number with $n$ prime.

It is an open problem if there are infinitely many Wieferich primes and infinitely many non-Wieferich primes.

In OEIS Wieferich numbers are A077816.

Comment in OEIS claims:

If there are finitely many Wieferich primes (A001220), this sequence is finite.

Let WP denote Wieferich prime and NWN denote non-Wieferich number.

It is unconditionally known that there are infinitely many non-Wieferich numbers with unbounded largest prime factor.

From now on assume there are only finitely many non-Wieferich primes and almost all primes are WP.

Let $p_i$ be distinct WPs. Then $n=\prod p_i$ is Wieferich number at it is not NWN.

Q1 Are there only finitely many squarefree NWNs?

Q2 What other properties NWNs have?

A guess:

Q3 Must the smallest prime factor of the NWNs be bounded?

Addedd Initially we asked about the largest prime factor, but Will Sawin disproved this and Q1 in a comment.

  • 1
    $\begingroup$ If $p$ is not a Wieferich prime, then any product of $p$ with primes not congruent to $1$ mod $p$ is not a Wieferich number. So the answers to Q1 and Q3 are negative. $\endgroup$
    – Will Sawin
    Nov 24 at 12:25
  • $\begingroup$ @WillSawin Many thanks, you are right. I edited Q3, asking about the smallest prime factor. $\endgroup$
    – joro
    Nov 24 at 14:08
  • 3
    $\begingroup$ Prime powers of any prime will eventually be NWN's, so I think the answer as stated to Q3 is still negative. If you mean squarefree then I think this is true: A product of distinct Wieferich primes is a Wieferich number, so the smallest prime factor of a squarefree non-Wieferich number must be at most the largest non-Wieferich prime. $\endgroup$
    – Will Sawin
    Nov 24 at 14:15
  • $\begingroup$ @WillSawin I am trying to exploit the fact that Mersenne numbers $M_n$ are NWNs, but the squarefree part is obstacle. Maybe the fixed Q3 will be elementary result about $M_n$ being squarefree. $\endgroup$
    – joro
    Nov 24 at 14:52


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