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On the square-freeness of Fermat and Mersenne numbers. (English) Zbl 0149.28204
Authors’ summary: It has been conjectured that the Fermat and Mersenne numbers are all square-free. In this note it is shown that if some Fermat or Mersenne number fails to be square-free, then for any prime $p$ whose square divides the appropriate number, it must be that $2^{p-1} \equiv 1\pmod{p^2}$. At present there are only two primes known which satisfy the above congruence. It is shown that neither of these two primes is a factor of any Fermat or Mersenne number.

MSC:
11A51Factorization; primality
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