I can show that it can only work for odd numbers :-). There are two directions to examine: a) Dig up Pinch's list of 2-pseudoprimes & try them. (Is there a bigger list?) b) Try using a base other than 2. We know 2^341=2 (mod 341) with 341=11*31. Find N from 2^341 = 2+341N (mod 341^2). Then look at (2+341N)^341 and expand the first terms of the binomial ... 2^341 + 341 * 2^340 * 341N + stuff*(341N)^2 so (2+341N)^341 = 2^341 = 2+341N (mod 341^2). This shows that the composite 341 has some base 2+341N that makes 341 a "Wieferich pseudoprime". If there are no base 2 WPSPs, it's probably just luck. Rich ----- Quoting Bill Gosper <billgosper@gmail.com>:

Trying a few more, In[6]:= Select[Range[10^9], PowerMod[2, #, #^2] == 2 &] // tim

During evaluation of In[6]:= 2031.078479,2

Out[6]= {1093, 3511} Rich? Anybody? Is it obvious this can only yield primes? Time to add the conjecture to A001220? --rwg

On Mon, Mar 12, 2018 at 12:36 PM, Bill Gosper <billgosper@gmail.com> wrote:

...

https://oeis.org/A077816 , e.g., In[19]:= Select[Range@9999, Mod[2^EulerPhi@#, #^2] == 1 &]

Out[19]= {1093, 3279, 3511, 7651}

A different definition might be In[18]:= Select[Range@999999, Mod[2^#, #^2] == 2 &]

Out[18]= {1093, 3511}

No mention of primes! https://oeis.org/A001220

Arthur Wieferich showed that if p is not a term of this sequence, then the First Case of Fermat's Last Theorem has no solution in x, y and z for prime exponent p (cf. Wieferich, 1909). - Felix Fröhlich <https://oeis.org/wiki/User:Felix_Fröhlich>, May 27 2016

--rwg

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