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., 2^p-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 = 2^4q+3-1 (4q+3 prime, 8q+7 composite), and we calculate the residues (modulo N) of the series

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.