CHAP. XXII] SUM OF SQUARES A BlQUADRATE. 667 



Examples for n = 4, , n = 7 are deduced. To proceed otherwise when 

 w = 3, employ the numbers 2mp, 2rp, m?-\-r' 2 p 2 . Their sum is the square 

 of mr+p if m=(p 2 2rp)/r. Then the sum of their squares equals 

 (ra 2 +r 2 +p 2 ) 2 and is a bi quadrate if 



whence p = 4r, m = 8r, and the desired numbers are 64r 2 , 8r 2 , 49r 2 . A. 

 Martin employed 2a,-s(i = 1, -, n 1), aj+ +fl_i s 2 as the n numbers 

 and wrote m = a 2 + +a n _i. Then shall 



Take s = A 5, 2ai+2?ft 2s = A+. Then either of the preceding equa- 

 tions gives a\. 



R. Goormaghtigh 299 discussed 



MISCELLANEOUS SYSTEMS OF EQUATIONS OF DEGREE FOUR. 



Diophantus, V, 5, found three squares such that the product of any two 

 added either to the sum of the same two or to the remaining one gives a 

 square (cf. Fermat 100 of Ch. XIX). 



J. Prestet 300 found three squares such that the product of any two added 

 to the product of a given square a 2 by either the sum of those two or the 

 remaining one gives a square. For a = 3, he found 25, 64, 196. 



Beha-Eddin 301 (1547-1622) included, among seven problems remaining 

 unsolved from former times, 



Prob. 1: x+y = W, (z+z 1/2 )(?/+?/ 1/2 )= given; 



Prob. 5: x+y = 10, -+- = x. 



y ^ 



Fermat 302 noted that z 4 ?/ 4 is a cube and xy=ltfx= 13/22, y= 9/22, 



while positive solutions can be found by setting x z-\- 13/22, yz 9/22. 



L. Euler 303 required three numbers x, y, z such that k=x 2 y 2 -\-x 2 -\-y z , 



+2/ 2 +z 2 shall be all squares. He took z 2 = x 2 -\-y 2 +l -\-2-Jk. For y = x+l 

 we have k = w z , z z = 4w, where w = x z +x+l. Now w=Hl = (t z) 2 for 

 x = (t 2 - 1)/(2<+ 1) . Then the solutions are 



_/-! _ 2 -f2 _2 2 +2 + 2 



X ~2t+l' y ~~2t+i' Z ~ 2t+l ' 



Euler 304 treated the three problems to make (i) AB and AC squares; 

 (ii) BC a square; (iii) B and C squares, where 



299 L'intermgdiaire des math., 25, 1918, 17- 18. 



300 Elemens des Math., Paris, 1675, 331. 



301 Essenz der Rechenkunst von Mohammed Beha-eddin ben Alhossain aus Amul, arabisch 

 u. deutsch von G. H. F. Nesselmann, Berlin, 1843, 55-6. French transl. by A. Marre, 

 Nouv. Ann. Math., 5, 1846, 313. Cf. A. Genocchi, Annali di Sc. Mat. e Fis., 6, 1855, 297. 



302 Oeuvres, I, 300-1; French transl., Ill, 248-9. Observation on Diophantus, IV, 12. 



SOB N ov i Comm. Acad. Petrop., 6, 1756, 85; Comm. Arith., I, 258; Op. Om., (1), II, 426. 



304 Novi Comm. Acad. Petrop., 20, 1775 (1771 ), 48; Comm. Arith., 1, 444; Op. Om., (1), III, 405. 



