On Certain Properties of Prime Numbers.
On Certain Properties of Prime Numbers.
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DOI：
10.1098/rspl.1843.0113
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通讯作者：
F. Pollock
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作者：
F. Pollock
本文的作者在注意到威尔逊定理（由 Waring 于 1770 年左右发表，没有任何证明）后，该定理是，如果 A 是素数，则 1. 2. 3. . 。 。 (A —1)+1 可被 A 整除；提到拉格朗日和欧拉的证明，并提到高斯将该定理扩展到任何数，但不包括素数；前提是不是 1、2、3 等。 (A —1)，只取与A互质的数，并加或减1。这个定理是由高斯于1801年在没有证明的情况下发表的，其中规定了加或减1的情况，其正确性受到作者的质疑，他接着提出了以下定理，他之前为了清楚起见将其分为三个。如果取任何一个数，无论素数与非素数，并且确定与它素数的数，且小于该数的二分之一，并且那些平方±1等于该素数或其倍数（可能大于一）的数被拒绝，则其余素数（如果有）的乘积±1应能被该素数整除。
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.