THEORY OF COMMENSURABLES. 729 



Taking the operation in detail, we have, on supposing 



A„+i, B„+i, 7"+!, &c. to have been deduced in succession from 

 a, b, 7, 

 we have 



A„+i=aA„ + &B„; B„+i=aB„-6A„ . 



If now we deduce from these, 



A„+2 = « A„+i + bBn+i ; B„+2 = aBn+i — 6 A„+i , 

 the results are, 



A„+2=(a'-*')A„ + 2a&B„; B„+2 = -2abA^+ (a^ -b^)Bn, 

 which, if A„ and B„ be prime to each other, have no common divisor. 

 But if we take the second combination, and put 



A^+2=«A„+i — 6B„+i ; B„+2=aB„+i + 6A„+i , 



we obtain 



A„+2= (a2 + b^)A„ ; B„+2 = (a^ + b')B„ ; 7"+^ , 



which are all divisible by a^ + If or 7^ and bring us back to 



A„, B«, 7". 



And thus we see that in forming the successive trigons A2, B2, y'^ ; A3, B.^, 7% 

 &c. from a,h^l\ and confining ourselves to those only which are in their lowest 

 terms, we form only one series. 



21. Every number which is the product of 72 primes of the form 4w+ 1, or of 

 any powers of those primes, may be the hypotenuse of 2"~^ right-angled trigons 

 in their lowest terms. 



In regard to the obtaining of trigons in their lowest terms, the power 7" gives 

 only one as 7 itself does ; wherefore the combinations 7" ^^ ^\ &c. give just as 

 many cases as the product 7^^, &c. 



General Scholium. 



From these propositions we obtain a convenient method of computing and 

 tabulating the cases of right-angled trigons in the lowest terms. 



We form a list of the prime numbers of the form An+l; these are 5, 13, 17, 

 29, 37, &c., and we decompose each of these into two squares ; thus, 2^-|- P = 5 ; 

 3^ + 2^ = 13; 4'^ = 1^ = 17, and so on, the general form being ^^'^ -I- ^'-^ = 7. From 

 these decompositions we then obtain, according to the formulae 



a=^2pq, b=p^ — q^, 



the sides of the trigons of which these prime numbers are the hypotenuses. 



Having thus constructed, to whatever extent we may desire, the table of 

 trigons with prime hypotenuses, we combine them according to the formulae of 

 articles 18, 19, 20, and obtain the cases in which the hypotenuses are composite. 



VOL. XXm. PART III. 9 K 



