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23. A NEW
RESULT CONCERNING A
MERSENNE NUMBER.—In
Scripta Mathematica, v. 3,
1935, p. 112-119,
there is an
article by R. C. ARCHIBALD on "Mersenne's
numbers," Mp
= 2p - 1,
(p prime). This article tabulates all results known in 1935
(except one) for these numbers p = 2 to p = 257.
Archibald has pointed out that the one result he had overlooked was
that announced by POWERS in London Math. So.,
Proc., v. 12, Mar. 13, 1913, p. iii, that
2257 - 1
has no factor < 10,017,000. Until recently it was not known whether
6 of the 55 Mersenne numbers were prime or composite. On August 11,
1944, I completed the proof that the smallest of these 6 numbers,
M157, is composite.
Details concerning my computations are about to appear in Nat. Acad.
Sci., Proc. Our present knowledge of the
Mp is tabulated below.
| p |
Character of
Mp |
| 2,3,5,7,13,17,19,31,61,89,107,127 | Prime |
| 11,23,29,37,41,43,47,53,59,67,71,73,79 |
Completely factored |
| 113,151,179,223,233,239,251 |
Incompletely factored but two or more prime factors known |
| 83,97,131,163,173,181,191,197,211 |
Only one prime factor known |
| 101,103,109,137,139,149,157,241,257 |
Composite but no prime factor known |
167,193,199,227,229 | Character unknown |
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