CHAP. IV] RIGHT TRIANGLES WITH A GIVEN DIFFERENCE OF LEGS. 183 



gives x = (2pq + 2q 2 }fd, where d = p 2 2q 2 . He made d = 1 by use 

 of the theory of Pell's equation. 



A. Martin 121 gave the nth triangle for n = 80 and 100. 



P. Bachmann 122 proved that the only integral solutions of x 2 + y 2 = z 2 

 in which z > 0, while x and y are consecutive integers, are those given by 



x + y + zJ2 = (1 + V2)-(3 + 2V2) fc (k = 0, 1,2, ) 



Several writers 123 obtained the first six triangles. 



R. W. D. Christie 124 noted that the solution of x 2 + (x + I) 2 = y 2 in 

 integers is 



x = 2 + 2i + + 2 2r _!, y = 2 2r , 



where 2 r is the simple continuant of order r all of whose diagonal elements 

 are 2. This was proved by T. Muir, 125 who cited Fermat's 110 rule. 



A. Martin 126 noted various methods. The first three methods are based 

 on the solution of 2k 2 1 = D [Ozanam, 112 Hutton, 113 Bachmann 122 ]. 

 Fermat's method was used to compute a table (p. 322) of the first forty 

 such triangles. 



A. LeVy 127 found when two of the numbers (1) are consecutive. Evi- 

 dently z y = 2n 2 4= 1. Next, z x = (m n) 2 = 1 for m = n + 1. 

 Finally, y - x =_ 1 gives (m - n) 2 - 2n 2 = 1. Write (1 - V2)p in 

 the form a b^2; then a, b are integral solutions of a 2 2b 2 = ( l) p , 

 and all solutions of u 2 2v 2 = db 1 are said to be obtained in this way by 

 using all integral values of p. Or we may compute the solutions of the 

 latter by the recursion formulas of G. Fontene 284 of Ch. XII. We get 



(3, 4, 5), (21, 20, 29), (119, 120, 169), (697, 696, 985), (4060, 4059, 5741). 



G. A. Osborne 128 discussed the problem. Cf. Barisien 100 of Ch. IX. 

 Several 129 made use of x 2 2y 2 = 1. F. Nicita 130 employed recurring series. 



RIGHT TRIANGLES THE DIFFERENCE d OR SUM OF WHOSE LEGS is GIVEN. 



Frenicle 131 stated that every number is the difference of the legs in an 

 infinitude of ways, every prime 8n + 1 or product of such primes is the 

 difference of the legs of an infinitude of primitive triangles. To find all 

 triangles with d = 7, start with (5, 12, 13) formed from 3, 2, and take that 

 formed from 3, 2-3 + 2, etc. A second series is found similarly from 

 (8, 15, 17), formed from 4, 1. He discussed right triangles the sum of 

 whose legs is given. 



121 The Analyst, Des Moines, 3, 1876, 47-50; Math. Visitor, 1, 1879, 56, 122 (erroneous values 



for n = 5, 6 occur on pp. 55-6). 



122 Zahlentheorie, 1, 1892, 194-6; Niedere Z., 2, 1910, 436. 



123 Amer. Math. Monthly, 4, 1897, 24-28. 

 1M Math. Gazette, 1, 1896-1900, 394. 



126 Proc. Roy. Soc. Edinburgh, 23, 1899-1901, 264-7. 



126 Math. Magazine, 2, 1910, 301-24. 



127 Bull, de math, elementaires, 15, 1909-10, 165-6. 



128 Amer. Math. Monthly, 21, 1914, 148-150. 



129 L'intermediaire des math., 22, 1915, 139-144, 185-8. 



130 Periodico di Math., 32, 1917, 200-210. 



131 Oeuvres de Fermat, II, 235-6, 238-41, letter to Fermat, Sept. 6, 1641. 



