CHAP, xix] SQUARES, SUM OF ANY Two LESS THIRD A SQUARE. 509 



y = 2mn 1. Then the first condition (A) is satisfied if 



n = m/z, z = m z 

 There remains the condition 



which is satisfied if m = p 2 -g 2 , p 4 +g 4 -4p 2 g 2 = (p--2rg 2 ) 2 , whence 



For r = 13, p = 15g/4 and the numbers are proportional to Legendre's. 



FURTHER SETS OF THREE OR MORE LINEAR FUNCTIONS OF THREE OR MORE 



SQUARES MADE SQUARES. 



Leonardo Pisano, 51 to make x~+y 2 , x 2 +y 2 +z 2 , x 9 -+y-+z-+w z , - all 

 squares, took the first square x- to be 9. Then the second, y 2 , is the sum 

 16 of all odd numbers 1, 3, 5, 7 preceding 9, whence 9+16= D = 25. As 

 the third square take the sum 144 of all odd numbers < 25 whence 

 144+25= D = 169. As the fourth square take 1+3H ----- [-167 = 7056 

 whence 7056+169= D =7225. As the fifth square take 



1+3H ---- +7223 = 13046444. 



Leonardo noted (p. 279) that, since 7225 is the square of 85, not a prime, 

 we can get several values for the fifth square. Besides that given above 

 we may take the sum of all odd numbers ^ 7225/5 5 1 and get the 

 square 720 2 , or the sum of all odd ^ 7225/25-25-1 and get 132 2 . A. 

 Genocchi 52 noted that a fourth solution was omitted, viz., the sum 204 2 

 of all odd =i 7225/17-17-1. 



F. Feliciano 53 gave only 9, 16, 144. 



N. Tartaglia 54 obtained 25, 144, 7056 by Leonardo's method. 



J. de Billy 55 found the squares 9, 1/100, (23/15) 2 such that if 15 is added 

 to the sum of any two of them there results a square. [Due to Diophantus, 

 V, 30; cf. Fermat 9 of Ch. XV.] 



L. Euler, 46 p. 604, stated that it is not possible to find four squares 

 such that if each be subtracted from the sum of the remaining three the 

 difference is always a square. 



H. Faure 56 proved the last theorem by use of the lemma that 

 2x~+2y z +2xy = z 2 is impossible in integers. 



Euler 57 noted five sets of solutions, like p = S9, 3 = 191, r = 329, of 



61 Scritti, II, 254, note on margin; 279. Tre Scritti, 57, 112. 



62 Annali di Sc. Mat. e Fis., 6, 1855, 355-6. 



63 Libro di Arith . . . Scala GrimaldeUi, Venice, 1526, f. 5. 



64 La Seconda Parte Gen. Trattato Numeri et Misure, Venice, 1556, f. 142 left. 

 66 Diophantvs Geometria, Paris, 1660, 117-8. 



66 Nouv. Ann. Math., 16, 1857, 342-4. 



67 Opera postuma, 1, 1862, 259-60 (about 1782). 



