665 



The author of this paper, after noticing Wilson's Theorem, (pub- 

 lished by Waring about the year 1770, without any proof), which 

 theorem is that, if A be a prime number, 1. 2. S. . . . (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 sub- 

 tracted. 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 sub- 

 tracted, the correctness of which is questioned hj the author, who 

 proceeds to propound the following theorem, which he had previ- 

 ousl}'-, 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 re- 

 maining primes (if any), + 1 shall be divisible by the prime number. 



He gives as examples, 14-, the primes to which, and less than one 

 half, ai-e 1, 3, 5, and 1. 3. 5 = 15; therefore 1. 3. 5 — 1 = 14; also 

 15, the primes to w^hich and less, are 1, 2, 4, 7; but 4x4 = 16 

 = 15 + 1 ; therefore 4 is to be rejected, and 1. 2. 7 + 1 = 15. The 

 author adds another theorem, that if A be a prime number, all the 

 odd numbers less than it (rejecting as before) ; also, all the even 

 numbers (making the same rejection except A — 1) will, multiplied 

 together, be equal to A + 1. 



The author then proceeds to prove Gauss's extension of Wilson's 

 theorem, and to give the cases in which 1 is to be added or sub- 

 tracted ; and in the course of the proof, he mentions that the num- 

 bers prime to any number not only are found in pairs, one greater 

 and one less than one-half of the number, but that they associate 

 themselves in sets of four, with an odd pair in certain cases. Thus, 

 the primes to 7 are 1, 2, 3, 4, 5, 6> — 



2x4 = 8 = 7 + 1. 



Put the complemental numbers underneath crosswise, thus, — . 



2x4 



3 X 5 



so that 2 + 5 and 4 + 3 may equal 7 ; and then 

 3x5=15=2x7 + 1 



2x3= 6=7-1 4x5=20=3x7-1 



Multiplied together one way the product exceeds 7, or a multiple 

 of it, by 1 ; multiplied the other way, the product is less than 7, or 

 some multiple of it, by 1. By assuming the prime number to be A, 

 and the two primes to it to be p, q, and that p + q he not equal to 

 A, but pq=nA+l, it is shown that the complemental primes 



