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MR EDWARD SANG ON THE 



hypotenuse of four right-angled trigons in integers ; of which two are in their 

 lowest terras, and two are reducible by the divisors 7, ^■ 



For, since every prime number of the form in + 1 is the sum of two squares, 

 Ave may put 



Y'=p^ + q^, 



8' = r' 



+ s' 



whence we can form the four equations 



y^d^ = (pr + qs)- + (ps — qry 

 y^d^ = (pr — qsy + {ps + qrf 



f8^ = 

 /^8^ = 





+ y-s^ 



+ 



8'-^q' 



giving the four trigons 



pr + qs, ps — qr, yo 



pr — qs, ps + qr, yd 



yr , ys , yd 



dp , 8q , yd 



of which the two first are in the lowest terms, while the two latter are reducible. 



19. The product of n primes, each of the form 4w+l, is the hypotenuse of 

 2»-i right-angled trigons in their lowest terms. 



For with one prime 7 we can have one trigon ; with the product of two primes, 

 7^, we can have two trigons. On introducing a new prime factor e, we can com- 

 bine the sides belonging to it with each of the two former trigons in two ways, 

 thus obtaining four cases. With a fourth factor ^, we can obtain two for each of 

 these four, and thus we proceed doubling the number of trigons at each accession 

 of a new factor. 



20. Every power of a prime number of the form in+l is the hypotenuse of 

 one sole right-angled trigon in its lowest terms. 



If a, b, 7, represent the three sides of a right-angled trigon ; that is, let 



a^ + b^ = y^. 

 Also put A, B, T, for those of a trigon having 7" for its hypotenuse, or, 



An' + -B,-' = y'\ 

 Multiply these two equations and we obtain 



Aja^ + A„262 + B^i^2 + B^j^2 ^ ^2^+2^ 



(A.a + B„&)2 + (A„b - B,ay = (7"+^)^ 



or, 



(A„a - B„b)2 + (A„6 + B„a)2 = (v-'+i)', 



from which it would appear that for every trigon with the hypotenuse 7" we 

 have two with the hypotenuse 7"^^ • One of these, however, is reducible, the 

 other is not. 



