MR H. F. TALBOT ON THE THEORY OF NUMBERS. 49 



If now M^e turn to primes of the form 47i + 3, the numbers less than ^? form 

 2n + l pairs, an odd number, .-. the product of each pair ^ — 1, and there are no 

 deviations. 



For example, if ^=7, the associates are 



12 4 



6 3 5 

 If p=ll, they are 



12 3 4 6 



10 5 7 8 9 



We will now pass to the consideration of another system of associate numbers, 

 which I do not find mentioned in the books. 



Theorem. — If ^ is a prime number of the form 4w+ 1, and the series of natural 



numbers 1, 2, 3, &c., be taken as far as ^^^ (which will be of the form 27i, and 



» — 1 

 therefore an even number), then the squares of these - ■ .. numbers can be divided 



into associate pairs, in such a wa}'^ that the sum of each pair shall be divisible hy p. 



« — 1 

 Example. — Let ^9 = 17, .*. —^ — =8. The 8 squares may be divided into pairs, 



so that each pair is divisible by 17, as follows: — 



12 + 42 22 + 82, 32 + 52, 62 + 72. 



It is plain that each number can have only one associate. For let a have the as- 

 sociate h .'. a^ + h^ ^0 (mod. pi). If c were another associate, we should have 

 «2 + c2 ^ (mod. p), and .-. Jy^-c^- ^ (mod, p)) ; that is, p must divide one of the 

 factors of &2_c2. But these are h + c and h—c. And J + c is less than p)^ because 



» — 1 

 h and c are each less than, or equal to, "^-q"- Much more is 5 — c less than p>. 



But p cannot divide numbers less than itself, therefore a has only the associate 

 h. It remains, however, to show, that each number has an associate. This follows 

 from the well-known theorem, — " That every prime of the form An + 1 is the sum 

 of 2 squares, in one way only." 



Sometimes one of the squares is unity. For example, the prime 17 is the 

 sum of 1 + 16 — 1^ + 4^ When this happens, the other associates are easily de- 

 duced. Thus, multiplying the equation 1^ + 4^ = 1 7 =p by 2^ we have 2^ + 8^ = 2^ . p, 

 which being divisible by p, is ^ (mod. p) .\2^ + & ^0, and 2 has the associate 



» — 1 

 8. Similarly, 3^ + 12^ ^ 0, but 12 exceeding -^ or 8, we substitute for it p>-\2, 



or 5, .'. 32 + 52 ^ 0, and 3 has the associate 5 ; and so on. 



But when the prime p is the sum of 2 squares, neither of which is 1, we 



proceed a little differently. Thus, let ^=29, which =4 + 25=22 + 5^ Multiply 



the least of these numbers, a, by the number which will give a product nearest to 



the prime 29, and the difference will of course be less than a. Thus, if we 



VOL. xxiii. PART I. 



