98. MERSENNE
NUMBERS.—In Nat. Acad. Sci., Proc.,
v. 34, Mar. 1948, p. 102-103, Professor
H. S. UHLER gives details of his proof
(completed 27 Nov. 1947) that
M193 is composite.
Thus he brought to a conclusion work begun in 1944
(MTAC, v. 1, p. 333) when the
characters of just six
of the Mp,
p = 157, 167, 193, 199, 227, 229, were unknown. He has now
shown that all of these are composite. See also MTAC,
v. 1, p. 404;
v. 2, p. 94,
341. Professor Uhler's final
summary of some of the facts concerning the 55 Mersenne numbers is
as follows:
| p | Character of
Mp |
| 2,3,5,7,13,17,19,31,61,89,107,127 | Prime |
| 11,23,29,37,41,43,47,53,59,67,71,73,79,113 | Composite
and fully factored |
| 151,163,173,179,181,223,233,239,251 | Two or more prime
factors known |
| 83,97,131,167,191,197,211,229 | Only one prime factor
known |
| 101,103,109,137,139,149,157,193,199,227,241,257 |
Composite but no factor known |
|