618 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxn 



area of the triangle formed from v 2 and w 2 would be a square. Again, 



x/y-y/x^O by (x 2 -y 2 )xy^O. 



J. Ozanam 12 stated that x 4 y 4 3=z 4 . For, a 4 b 4 is the area of the right 

 triangle whose sides are the ratios of 2a 2 6 2 , a 4 6 4 , a 4 +6 4 to ab, and is not a 

 square " as proved by Messieurs de TAcad. Roy. Sc. and also by R. P. 

 de Billy." 



L. Euler 13 proved that a 4 +6 4 + D if ab^O. For, if (a 2 ) 2 +(& 2 ) 2 = D, 

 where a and 6 are relatively prime, then a 2 p 2 q 2 , b 2 = 2pq, where p and q 

 are relatively prime, one even and the other odd. By p 1 q 2 = D, p is odd, 

 whence q is even. By p(2q)=b 2 , p and 2q are squares. By p 2 = a 2 +# 2 , 

 we get p = m?-}-ri 2 , q = 2mn, m and n relatively prime. Since 2#=D, 

 mn= D and m = x 2 , n = y 2 . Thus x 4 +y 4 is a square p, and x, y are less than 

 a, b. By a similar proof, a 4 b 4 H= D unless b = or b = a. 



E. Waring 14 and A. M. Legendre 15 reproduced literally these proofs by 

 Euler. 



C. F. Kausler 16 treated x 4 +y 4 = z 4 by use of the lemma that # 2 i/ 2 are 

 not both squares. Equating x = 2PQ, y = P 2 Q 2 (from x 2 -\-y 2 =\3, x, y 

 relatively prime) to x, y = p 2 +q 2 , p 2 q 2 or to (p 2 +q 2 )/2, (p 2 g 2 )/2, in either 

 order, where p and q are relatively prime, and odd in the latter case, we 

 are led to a contradiction. Now x 4 = z 4 y 4 requires z 2 -}-y 2 , z 2 y 2 = m 4 n 4 , I 

 or m 3 n 4 , m, (19 cases); 7 cases are excluded by the lemma, others by 

 z 2 +2/ 2 >2 2 y 2 or (z 2 y 2 ) z >z 2 -{-y 2 . Finally, if z 2 +y 2 = m s , z 2 y- = mn 4 , then 

 2z 2 = m(m 2 +n 4 ), while m = 2 is easily excluded. Thus [a prime factor of] 

 m is a factor of z and hence of y. 



P. Barlow 17 noted that, if the area xy/2 of a right triangle (a;, y, z) were 

 a square w 2 , then 2 2 db4w 2 =(z?/) 2 , whereas it was proved by descent 

 (p. 109) that x 2 +y 2 and x 2 y 2 are not both squares. Also (p. 119), 



J. Horner 18 noted that if x/yyfx= D, where x, y are relatively prime, 

 then x = m 2 , y = n z , m 4 n 4 = D, contrary to a known result. 



Schopis 19 proved x*-\-y 4 = z 2 impossible, using the impossibility of 

 x 4 y 4 = 2z 2 . Next (pp. 6-10), x*+y 4 = 2z 2 is impossible; likewise (p. 11) 

 x 4 -y 4 = z 2 . 



A. M. Legendre 20 stated that the above 3 proof that the area of a right 

 triangle is not a square shows that a 4 & 4 + D if a=^b, 6 + 0. [But in 



12 Journal des Sgavans, 1680, p. 85. 



I3 Comm. Acad. Petrop., 10, 1747 (1738), 125-34; Comra. Arith., I, 24-34; Opera omnia, 

 (1), II, 38. Same proofs in Euler's Algebra> 2, Ch. 13, arts. 202-8, St. Petersburg, 

 1770, p. 418; French transl., Lyon, 2, 1774, pp. 242-54; Opera omnia, (1), 1, 1911, 437; 

 Sphinx-Oedipe, 1908-9, 59-64. 



14 Meditationes Algebraicac, Cambridge, ed. 3, 1782, 371-2. 

 "Theorie des nombres, Paris, 1798, 404; cd. 2, 1808, 343; cd. 3, 1830, II, 5; German 



transl. by Maser, 2, 1893, 5. 



Nova Acta Acad. Petrop., 13, ad annos 1795-6 (1827), Mem., 237-44. 

 Theory of Numbers, 1811, 121 (cf. 144). 



8 The Gentleman's Diary, or the Math. Repository, London, No. 80, 1820, 37. 

 9 Einigc Siitzc aus der unbestimmten Analytik, Progr. Gumbinnen, 1825. 

 Th6orie des nombres, ed. 3, 2, 1830, 325, p. 4, Cor. (Maser, II, p. 4). 



