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Recall that a Wieferich prime is a prime number $p$ such that $2^{p-1} \equiv 1 \bmod p^2.$ It is not known whether there are infinitely many Wieferich primes, nor whether there are infinitely many non-Wieferich primes. In fact there are only $2$ known Wieferich primes.

I'm interested in a slightly different condition which I'm hoping is easier to handle. Namely, I can replace the exponent $p-1$ by the order of $2$ in $(\mathbb{Z}/p\mathbb{Z})^\times$. Moreover, I just want this power to hit the identity with odd $p$-adic valuation. Specifically:

Are there infinitely many primes $p$ such that $v_p(2^{\mathrm{ord}_p(2)}-1)$ is odd?

Here $\mathrm{ord}_p(n)$ denotes the order of $n$ in $(\mathbb{Z}/p\mathbb{Z})^\times$ (which divides $p-1$ by FLT), and $v_p$ is the $p$-adic valuation.

Note that the existence of infinitely many non-Wieferich primes would provide a positive answer to my question (since here $v_p(2^{p-1}-1) = 1$).

Ideally I'd also like to know that there collection of such primes has positive density, rather than just being infinite.

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  • $\begingroup$ Do you possibly mean $v_p(2^{{\rm ord}_p(2)}-1)$ instead of $v_p(2^{{\rm ord}_p(2)})$? $\endgroup$ – Arno Fehm Jun 29 '20 at 20:37
  • $\begingroup$ Yes thanks, now fixed. $\endgroup$ – Daniel Loughran Jun 29 '20 at 20:44
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    $\begingroup$ Since $v_p(2^{{\rm ord}_p(2)}-1)=v_p(2^{p-1}-1)$, you might as well keep the exponent simpler. $\endgroup$ – Pace Nielsen Jun 29 '20 at 21:59
  • $\begingroup$ @Pace: Nice observation. How do you show this? $\endgroup$ – Daniel Loughran Jun 30 '20 at 10:32
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    $\begingroup$ @DanielLoughran Let $x={\rm ord}_p(2)$. Write $2^x=1+ap^k$ where $gcd(p,a)=1$. Now, the order of $2$ modulo any larger power of $p$ must be a multiple of $x$, say $xy$. We compute that $2^{xy}=(1+ap^k)^y=1+yap^k+$terms divisible by $p^{k+1}$. We see that this is $1$ modulo $p^{k+1}$ if and only if $p|y$. $\endgroup$ – Pace Nielsen Jun 30 '20 at 16:24
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Suppose the answer is no and that the finitely many exceptions are all at most $B$. Let $\ell \equiv 1 \pmod{3}$ be prime and consider $n=2^{\ell} -1$. If $p>B$ is a factor of $n$, then $\ell$ is the order of $2$ modulo $p$, so $p$ occurs in $n$ with an even exponent, so $n = x^2c, c \le B!$. Let $y = 2^{(\ell - 1)/3}$. Then $n=2y^3 - 1$ and finally $2y^3 - 1 = cx^2$, so $(x,y)$ is an integral point on one of a finite collection of elliptic curves and there can be only finitely many such. But there are infinitely choices for $\ell$, contradiction. (This is a variant of an old argument of Granville.)

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  • $\begingroup$ This is a very slick argument! However it doesn't seem to get positive density, which I was originally hoping for. Which paper of Granville does this appear in? $\endgroup$ – Daniel Loughran Jun 30 '20 at 10:35
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    $\begingroup$ I think one can make the argument a bit simpler, keeping the same basic idea. Since $\ell$ is the order of $2$ mod $p$, we have $p > \ell$. Thus, as long as we choose $\ell > B$, we have $2^{\ell}-1$ coprime to $B!$. So $n$ is actually a square, which is absurd since $n\equiv 3\pmod{4}$. In this variant there's no need to restrict the primes $\ell$ to the progression $\ell\equiv 1\pmod{3}$, $\endgroup$ – so-called friend Don Jun 30 '20 at 19:01
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    $\begingroup$ I'm guessing the Granville paper in question is mathscinet.ams.org/mathscinet-getitem?mr=841645 . It seems that the state of the art only provides $\log X$ or so non-Weiferich primes up to $X$, even if one assumes the abc conjecture (see e.g. the recent paper mathscinet.ams.org/mathscinet-getitem?mr=3983273 ) so I doubt positive density is within reach of current methods. $\endgroup$ – Terry Tao Jun 30 '20 at 19:21
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    $\begingroup$ ... though on the other hand, this argument does seem to imply that for every prime $\ell$, there is a prime $p$ with $ord_p(2)=\ell$ and $v_p( 2^{ord_p(2)}-1)$ odd (because $2^\ell-1$ is not a square). Don't know if this is actually helpful for your application. $\endgroup$ – Terry Tao Jul 1 '20 at 0:38
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    $\begingroup$ @Terry Tao: This is a nice observation which could certainly be useful! It seems to give a more elementary proof of infinitude and also gives a stronger conclusion. $\endgroup$ – Daniel Loughran Jul 1 '20 at 15:53
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Felipe refers to my first ever paper (mathscinet.ams.org/mathscinet-getitem?mr=789713) from 1985 ! However I have a more recent paper that gives a better result along the lines asked for (mathscinet.ams.org/mathscinet-getitem?mr=2997580) which shows that every $2^n-1$, with $n\ne1$ or $6$, has a primitive prime factor that divides it to an odd power.

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  • $\begingroup$ Thanks Andrew! I'll take a look at your more recent paper. $\endgroup$ – Daniel Loughran Jul 6 '20 at 9:50

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