50 MR H. r. TALBOT ON THE THEORY OF NUMBERS. 



multiply the congruence 2^ + 5^ ^ by 15^ we get 30^ + 75^ ^ 0, and rejecting the 

 multiples of 29, we get 1 + (- 12)^ = 0, or 1 + 12^ = (because 75= 3 x 29 - 12), 

 And upon trial it will be found that 1 + 12^ or 145, is divisible by 29. Having 

 thus found a pair of squares, such as 1 + cv^ ^ 0, we find all the others from it by 

 simple multiplication, and rejecting the multiples of 29. If we had not found 

 this pair 1 +«"^ at first, we should at any rate have approximated to it. 



Another mode is the following : — Since 2^ + 5^=29=/? and 5 is not divisible 

 by 2, add 29 to it .-. 2' + 34^ =j9 = (mod. ])\ and dividing by 2", P + 17-^ = 

 .-. 1 + ( - 12)^ = 0, .'. 1 + 12' = 0, as before. 



p being a prime of the form 4?i + 1, we have in all cases j^ = wi2 + ?r, and having 

 ascertained the values of m^ n, we can derive from them other numbers «, 6, c, 

 d, such that a- + 6" ^ 0, d^ + d' ^ 0, &c. ; from which a curious theorem arises,— 

 If the prime number p divides both a^ + 5^ and c^ + d''\ it also divides both ac-^-bd 

 and he— ad. 



Eosam.ple.—2Q divides 22 + 5^ and 3' + 7". Therefore, it divides 57- 2-3 and 

 2-7 + 5-3. 



Demonstration. — Because a'^ + Jf^O and c^ + c?^ ^ 0, .• . a^c- + h^c^ ^ and 

 a^c- + a^d"^ ^0, .-.by subtraction Irc^ — a^d^ ^ 0, the factors of which being he + ad 

 and he- ad., p must divide one of them. [M.] 



Permute the letters a, h, in this result, since it is immaterial which is which ; 

 therefore p divides one of the two factors, ac + hd, or ae-hd. [N.] 



Comparing the results M and N, we see that if p divides ac + hd in the second 

 of them, it divides he— ad in the first. 



It appears from what precedes, that a prime p of the form 4y^ + 1 always 



divides some number of the form l + «^ where a is less than^^. Annexed is 

 a table of the values of a for the first prime numbers of that form, from 5 to 109. 



5 divides 1 + 2^ 



13 . 



. 1+52 



17 . 



. 1+42 



29 . 



. 1 + 122 



37 . 



. 1+62 



41 . 



. 1+92 



53 .. 



. 1 + 232 



61 divides 1 + II2 



73 . 



. 1 + 272 



89 . 



. 1 + 342 



97 . 



. 1 + 222 



101 . 



. 1 + 102 



109 . 



.. 1+332 



The law which governs these results is not manifest, therefore, althouo-h the 



prime p always divides a number of the form l+x"- (x less than ^~^ ) , yet x 



must be found by tentative methods. 



We will here add a few more examples of a theorem previously mentioned : 



The prime 13 = 22 + 3^ and divides 1 + 5- .'. 3-5- 1-2 = 13, and 2-5 + 1-3 = 13. 

 The prime 41=4^ + 5' and divides 1 +9^ .-. 4-9 + l-5=41, and 5-9-l-4=41. 

 The prime 61 = 52 + 6' and divides I + II2 .-. 5-11 + 1-6 = 61, and 6-ll-l-5 = 61. 



