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



J. Gediking 42 noted that, for relatively prime solutions of x z y 2 = z 2 , 

 we may take as x y any number of the form (2n + I) 2 or 2n 2 , but no 

 other. Then x + y = (2m + I) 2 or 2m 2 , with 2m + 1 and 2n + 1 or 

 m and n relatively prime. [It was overlooked that we may restrict to 

 one of the two cases.] All solutions < 1000 are given. J. C. Milborn 

 (pp. 167-9) erred in saying that this method does not give all solutions. 

 T. Boelen (pp. 238-40) noted that we may take as z any integer > 2, if 

 solutions with a common factor are allowed. 



C. J. van der Burg 43 gave an incomplete proof of (1). 



Fitting 44 discussed the relatively prime solutions of x 1 + y 2 z 2 by 

 setting z = x + a, whence y 2 = a(2x + a). Without loss of generality we 

 may take a to be an odd square 1, 9, 25, , and equate 2x + a to the 

 successive odd squares. 



W. Kluge 45 noted that x 2 + y 2 = z 2 is satisfied by 



x 2 = d{, d<x, y = s_A } g = tA 9 



and gave recursion formulas for computing successive solutions. 



E. Meyer 46 noted that Stifel's 16 formulas for diametral numbers do not 

 give all, for example not 33 56, and that he should have used 



a : b = m 2 n 2 : 2mn. 



He compared many known ways of solving x 2 + y 2 = z 2 . 



P. Lambert 47 solved x 2 + y 2 = z 2 by use of numbers a + bi> 



N. Gennimatas 48 would solve x 2 + y 2 = or by setting 2a = c + d, where 

 cd is a square x 2 , whence y = a d. 



*E. Haentzschel 48 " noted that from one rational right triangle we can 

 derive an infinity by use of the formulas for sin no. and cos na [cf . Vieta, 4 

 Ch. VI]. From two right triangles whose hypotenuses are primes of the 

 form 4/c-fl, we can derive an infinity by use of the addition theorem for 

 sine and cosine. By means of these theorems we can arrange in order the 

 proper solutions of x 2 -\-y 2 = z 2 . 



P. Quintili 49 attributed to F. Klein (!) the solution (1) of x 2 + y 2 = z 2 . 



A. E. Jones 50 discussed right triangles whose three sides are of the form 

 x 2 - 1. 



C. A. Laisant 51 noted that MQ, 2PN, P 2 + N 2 are sides of a right tri- 

 angle if M, N, P, Q are four consecutive terms of Fibonacci's series (Vol. I, 

 Ch. XVII of this History), so that P = M + N, Q = N + P. 



42 Vriend der Wiskunde, 25, 1910, 86-96. 

 t3 Ibid.,2G, 1911, 188-191. 



44 L'interme'diaire des math., 18, 1911, 87-90 (233-4). 



46 Verhandlungen der Versamm. deutscher Philologen u. Schulmanner, Leipzig, 51, 1911, 137. 

 Unterrichtsblatter Math. Naturwiss., Berlin, 19, 1913, 11. 



46 Zeitschrift Math. Naturw. Unterricht, 43, 1912, 281-7. 



47 Nouv. Ann. Math., (4), 12, 1912, 408-421. 



48 Zeitschr. Math. Naturw. Unterricht, 44, 1913, 14-15. 



480 Blatter fur d. Fortbildung d. Lehrers u. Leherin, Berlin, 6, 1913, 395-6. 

 49 II Boll. Mat. Sc. Fis. Nat., 16, 1915, 69-71. 



60 Math. Quest, and Solutions (contin. of Math. Quest. Educ. Times), 2, 1916, 18. 



61 Comptes Rendus des Sc. Soc. Math. France, 1917, 18-19. 



