172 HISTORY OF THE THEORY OF NUMBERS. [CHAP. IV 



NUMBER OF RIGHT TRIANGLES WITH A GIVEN SIDE. 



Report has been given above of the papers by Volpicelli, 26 Hart 33 

 and de Jonquieres. 29 See Fermat 10 and Frenicle 17 ^of Ch. VI and papers 

 19-32 of Ch. XIII. 



F. Gauss 58 noted that to every hypotenuse composed of k distinct 

 primes belong 



different pairs of legs, where [x~\ is the largest integer =i x. The legs are 

 relatively prime for 2 k ~ 1 pairs. 59 



D. N. Lehmer 60 proved that the number N of right triangles whose 

 sides are integers with no common divisor, and whose hypotenuse is ^ n, 

 is asymptotically n/(2ir). But, if the sum of the three sides is ^ n, 

 N = n (log 2)/?r 2 , asymptotically. 



O. Meissner 61 stated that the number P of integral right triangles with 

 one leg x = 2 m p 1 p" n (p's distinct primes) is: 



P = " 



(2P 2 + 1), P. s | { fi (2m, + 1) -l), 



Z I v=i J 



where [a] is the largest integer ^ a. Also P + 1 is the number of sets of 

 positive integral solutions z, y of z 2 y 2 = x 2 (x given). 



E. Bahier 62 noted that if A, B, , P are distinct odd primes the number 

 of right triangles one of whose legs is A*B* P* is 



+ 2 k ~ 1 af3y - IT. 

 If A = 2, we have only to replace a by a 1 in the last result. 



RIGHT TRIANGLES OF EQUAL AREA. 



Diophantus, V, 8, required three rational right triangles of equal areas. 

 If, .as in V, 7, ab + a 2 + 6 2 = c 2 , the right triangles formed 7 from c,a; c, b; 

 c, a + b have the same area abc(a + b). The chosen example has a = 3, 

 b = 5, c = 7. This solution was given in general form by F. Vieta, 

 Zetetica, IV, 11. Fermat 62a observed that if z is the hypotenuse and b, d 

 the legs of a rational right triangle, we obtain a new right triangle of the 

 same area by forming the triangle from z 2 , 2bd and dividing its sides by 

 2z(6 2 d 2 ). From this new triangle we may derive similarly a third, etc. 

 Apart from notation, this method is the same as the " construction' in 



68 tiber die Pythag. Zahlen, Progr. Bunzlau, 1894, p. 15. 



69 If the hypotenuse is 65, the legs are 25, 60; 16, 63; 33, 56; or 39, 52. 



60 Amer. Jour. Math., 22, 1900, 327-8. 



61 Archiv Math. Phys., (3), 8, 1904, 181. 



62 Recherche M6thodique et Propri6t6s des Triangles Rectangles en Nombres Entiers, Paris, 



1916, 21-27. 

 62a Oeuvres, III, 254-5; S. Fermat's Diophanti Alex. Arith., 1670, 220. 



