On Mersenne's Numbers: Mr. R. E. Powers.
The purpose of Mr. Power's paper is to show
that the Mersenne number
2^{107}1 is prime, four
mistakes having now been found in Mersenne's classification,
viz., 2^{p}1 proved
composite for p = 67, and prime for p = 61,
89, and 107, contrary to his assertion. That
2^{107}1 is a prime number
is shown by means of the following theorem, which was proved by Lucas
in 1878:—
If N =
2^{4q+3}1
(4q+3 prime, 8q+7 composite),
and we calculate the residues (modulo N) of the
series
3, 7, 47, 2207, ...,
each term of which is equal to the square of the preceding,
diminished by two units: the number N is prime if the residue
0 occurs between the (2q+1)th and the
(4q+2)th term; N is composite if none of the first
(4q+2) residue is 0.
The 106th term of the above series is congruent
to 0 (modulo 2^{107}1),
consequently the latter is a prime number.
