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On the square-freeness
of Fermat and Mersenne numbers
Leroy J. Warren and Henry Gilbert Bray |
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Vol. 22 (1967), No. 3, 563–564
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Abstract |
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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 2p−1 ≡ 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
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Milestones
Published: 1 September 1967
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Authors |
| Leroy J. Warren | |
| Henry Gilbert Bray | |
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