CHAP, xxii] AREA RIGHT TRIANGLE =f= D, x* + if 4= D. 619 



that proof it was known that a and b arc not both odd, a criticism due to 

 A. Genocchi 21 ]. 



J. A. Grunert 22 reproduced Euler's proof that a 4 +6 4 =}=c 2 . 



0. Terquem 23 proved by descent that x 4 y 4 = z~ is impossible. 



J. Bertrand 23 " proved that 4 +?/ 4 +z 2 somewhat as had Euler. 



P. Volpicelli 24 proved that no congruent number is a square. . For, if 

 pq(p"-q 2 )=a z , then h~=(p 3 q+pq^ 2 = a 4 +4p 4 q 4 , (a 4 

 whereas a difference of two biquadrates is not a square. 



V. A. Lebesgue 25 proved the impossibility of x*+y* = z 2 by descent. It 

 suffices to treat (2 a p) 4 +7/ 4 = 2 2 , where p, y, z are all odd, and y, z are rela- 

 tively prime. The factors zif of (2 a p) 4 have no common factor other 

 than 2. Hence 



22/ 2 = 2< 4 , z=F?/ 2 = 2 4a - 1 w 4 , p = tu, y = t*-2**- 2 u*. 

 The lower sign is inadmissible. Hence t 4 y 2 = 2' la ~-u 4 . Thus 



T. Pepin 26 proved the impossibility of x 4 y 4 = z 2 in integers 



W. L. A. Tafelmacher 27 proved the impossibility of x 4 +y 4 = z 4 . 



D. Gambioli 28 proved that x 4 y 4 = z~ is impossible in integers +0. 



T. R. Bendz 29 proved by descent from x 4 +4y 4 = z 2 that the area of a 

 right triangle is not a square. 



L. Kronecker 30 amplified Euler's 13 proof. 



G. B. M. Zerr 31 employed unproved assumptions in an attempt to prove 

 that the area of no right triangle is a square. 



A. Bang 32 noted that relatively prime solutions of x 4 z 4 = y 4 imply 



Thus y\-yl = 4:yl, so that 



2/i+2/2 = 2tti, y! 

 Hence u\-u^ = 2 u u\\ so that 



and Ua = ViV z v a Vi. By the third and fourth, U 2 l +u i 2 = 2vl 2 +2 21 vf. Then by 

 the second, 



21 Annali di Sc. Mat. e Fis., 6, 1855, 316, foot-note. His like criticism of the proof by Ter- 



quem 23 is not valid. 

 22 Klugel's Math. Worterbuch, 5, 1831, 1143. 



23 Nouv. Ann. Math., 5, 1846, 71-4. 

 23 Traite elem. d'algebre, 1851, 224-7. 



24 Atti Accad. Pont. Nuovi Lincei, 6, 1852-3, 89-90. 



25 Exercices d'analyse num^r., 1859, 83-4; Introd. a la theorie des nombres, 1862, 71-3. 



26 Atti Accad. Pont, Nuovi Lincei, 36, 1882-3, 35-36. 



27 Anales de la Universidad de Chile, 84, 1893, 307-320. 



28 Periodico di Mat., 16, 1901, 149-150. 



29 Ofver diophantiska ekvationen x n +y n =z n , DIBS. Upsala, 1901, 5-9. 



30 Vorlesungen iiber Zahlentheorie, 1, 1901, 35-8. 



31 Amer. Math. Monthly, 9, 1902, 202. 



32 Nyt Tidsskrift for Matematik, 16, B, 1905, 35-36. 



