On certain properties of prime numbers

Abstract

Abstract The author of this paper, after noticing Wilson’s Theorem, (pub­lished by Waring about the year 1770, without any proof), which theorem is that, if A be a prime number, 1. 2. 3. . . . (A —1)+1 is divisible by A ; refers to Lagrange’s and Euler’s demonstrations, and mentions Gauss’s extension of the theorem, to any number, not prime; provided that instead of 1, 2, 3, &c. (A —1), those numbers only be taken which are prime to A, and 1 be either added or sub­tracted. This theorem was published by Gauss without a proof in 1801, with a rule as to the cases in which 1 is to be added or sub­tracted, the correctness of which is questioned by the author, who proceeds to propound the following theorem, which he had previ­ously, for distinctness, divided into three. If any number, prime or not, be taken, and the numbers prime to it, and less than one half of it be ascertained, and those be rejected whose squares ±1 are equal to the prime number, or some multiple of it (which may be more than one), then the product of the re­maining primes (if any), ±1 shall be divisible by the prime number.