On Mersenne's Numbers: Mr. R. E. Powers
The purpose of Mr. Powers'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) residues is 0.
The 106th term of the above series is congruent to 0 (modulo 2^107-1),
consequently the latter is a prime number.