48 ME H. F. TALBOT ON THE THEORY OF NUMBERS. 



We will take the same example as before, the number 13. The associate 

 numbers are 1, 12 ... 2, 6 ... 3, 4 ... 7, 11 and 9, 10, the product of each 

 pair beings — 1 (mod. 13). Thus, for example, 7x11 = 77. Rejecting 78, 

 a multiple of 13, there remains —1. But the remaining pair of numbers, 

 5 and 8, produce the product 40, which, rejecting 39, a multiple of 13, is equiva- 

 lent to 1. Therefore 5x8^1 (mod. 13). It will be observed that the num- 

 bers have different associates in Euler's system and in this system, 2 being 

 associated Avith 6, and not with 7, &c. ; except that 1 is still associated with 

 12, and 5 with 8. 



I will add some other examples of this new system of associate numbers. 



If the prime number be 5, the associates are 1, 4, whose product ^ — 1, and 

 2, 3 whose product ^ + 1. This prime is of the form 4?z + 1, therefore the num- 

 bers less than it form 2n pairs, an even number ; therefore the product of one pair 

 deviates from the rest, as was observed before. Other examples of this, in primes 

 of the form An + \, are, 7?= 13. This case has been given before. The associates 

 are written one over the other in the following table, and the deviating pair 

 stands by itself : — 



1 2 



3 



7 



9 



r 



) 



12 6 



4 



11 



10 



i 



i 



) = 17 we find,- 



— 











12 3 



5 



6 



7 



9 



4 



16 8 11 



10 



14 



12 



15 



13 



The sum of the deviant pair is always equal to the prime number. Thus, 



4 + 13 = 17. It is worth remark, that the same holds in Euler's system, where 



the deviant pair are always 1 and p — \^ whose sum =j9. 



It will make the nature of these associate numbers plainer, if we subtract p 



»— 1 

 from each of those which exceed ^-k — • The remainders will be negative num- 



» — 1 

 bers, less than^^-^-- Thus, if j9=17, writing the associates one above the other, 



and their product in the lowest line. 



or 



1 



2 



-2 



3 



6 



5 



7 



4 



-1 



8 



-8 



-6 



-3 



-7 



-5 



-4 



-1 



16 



16 



-18 



-18 



-35 



-35 



-16 



-1 



-1 



-1 



-1 



-1 



-1 



-1 



+ 1 



Rule to find the pair of numbers which deviate from the rest. Find the number 



» — 1 

 oc less than ^ ,^ , such that 1+x^ is divisible by ^, which can always easily be 



done, and has only one solution. Then x and -a; are the pair required. 



