On Mersenne's Numbers: Mr. R. E. Powers.

    The purpose of Mr. Power's paper is to show that the Mersenne number 2107-1 is prime, four mistakes having now been found in Mersenne's classification, viz., 2p-1 proved composite for p = 67, and prime for p = 61, 89, and 107, contrary to his assertion. That 2107-1 is a prime number is shown by means of the following theorem, which was proved by Lucas in 1878:—

    If N = 24q+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 2107-1), consequently the latter is a prime number.

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