MR EDWARD SANG ON THE THEORY OF COMMENSURABLES. 723 



one. Omitting tlie case A x A, which cannot give a trigon, it follows that A^ may 

 be represented as the product of two factors in | {(1 + 2p) (1 + 2^) (1 + 2r) . . . — If 

 ways. Each one of these gives a distinct solution, for if A^ = P . Q be one of 

 these decompositions, 



A = ^/P:Q; B=J(P-Q); C=i(P + Q). 



Of the total number of these solutions, only those are in their lowest terms in 

 which P and Q are prime to each other. Therefore, if n be the number of sepa- 

 rate primes which enter into A as factor, the number of cases in their lowest 

 terms is 2""S and is irrespective altogether of the exponents y, q, r, &c. 



5. No double of an odd number can be the side of a rational right-angled tri- 

 gon in its lowest terms. 



For if A be the double of an odd number itself prime, or the product of two 

 or more prime factors, a, ^, its square is 4 a^ /3^ which can be resolved into un- 

 equal factors prime to each other only of the general forms 4 a^/9^ x 1 ; 4 a^ x /3^ 

 one of which is even and the other odd ; wherefore the suras and the differences 

 of these factors are all odd, so that in the solutions 



A=2af3; B = i(4a2/32-l); G=^ (ia^^^ + 1) 

 A=2a(3 ; B=i (4a2-/32) ; C=| (Aa^ + ^^) &c. 



the fraction ^ must occur. To remove this fraction we must double all, and then 



A=4a/3; B=4a2^2_i. G=ia^(3^-i-l : 

 A=4.a^; B=4a2-jS2 ; C=4a2 + ^2 : Scc. 



6. Any number of the general form 2' A// in which t exceeds unit, A and /j. being 

 odd numbers prime to each other, may be the side of a rational right-angled 

 trigon in its lowest terms. 



For A^ = 2^' X^ ij? and may be resolved into the two factors P = 2^'~^ A^ ; Q = 2/x^ 

 whence A = 2*Aju; B = 2''-' A'— /x'; C = 2''-' A' + iu', which are all integer and 

 prime to each other. 



If n be the number of odd primes which enter into A as factors, the total num- 

 ber of cases in their lowest terms is 2"~\ 



7. I and m being any two numbers whatever, P-i{-m\ 2lm and P—m^ repre- 

 sent the three sides of a right-angled trigon. 



For if A = 2lm, B = F-rre and G = P + m\ we have A' -t- B' = C^ 

 If I and m have a common divisor. A, B, C, can all be divided by the square 

 of that divisor. If / and m be prime to each other and both odd^ the numbers in 

 the above solution may be halved ; but if one be odd, and the other even, the solu- 

 tion as given is in its lowest terms. 



8. Of a right-angled trigon in integers, one of the two sides is divisible by 3. 

 In this investigation we may suppose that the trigon is in its lowest terms. 

 In the equation «^ + ^^ = c^ if the number a be not divisible by 3, it must be 



of one of the forms ?)n+l, ^n—\, so that its square must be of the form 3w-t- 1. 



