Vol. 22, No. 3, 1967

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On the square-freeness of Fermat and Mersenne numbers

Leroy J. Warren and Henry Gilbert Bray

Vol. 22 (1967), No. 3, 563–564

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 2p1 1( mod p2). 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.

Mathematical Subject Classification
Primary: 10.07
Published: 1 September 1967
Leroy J. Warren
Henry Gilbert Bray