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 = 2^p-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 2^257-1 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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