488 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xvn 



/, g relatively prime integers ; the upper sign is excluded by use of modulus 4. 

 Restricting to positive integers, we have u = a(3, f=or, g = fi 2 . Thus 

 (a 2 -h2j3 2 ) 2 = l+2(2/3 2 ) 2 , the discussion of which as a Pell equation leads to the 

 condition m 4 2n 4 = l. The upper sign is excluded as it leads to 

 l+c 4 = d 2 . For the lower sign, m? = n 2 = 1, as noted in his next paper (ibid., 

 p. 35). 



A. Genocchi 22 treated 2 2 = ?/ 4 +(?/ 2 I) 2 , obtained by eliminating x. For 

 y odd, y 2 1 = (/ 2 </ 2 )/2, y 2 =fg, where /, g are odd and relatively prime. 

 Thus/=m 2 , g = n 2 and 2(ra 4 +l) = (ra 2 +n 2 ) 2 , whence ra 2 = n 2 = l, x = l, by 

 known theorems on 2r 4 +2s 4 = D. For y even, y 2 l=f 2 g 2 , y 2 = 2fg, where 

 /, g are relatively prime, and / is even. Thus /= 2a 2 , g = /3 2 and p 2 1 +8a 4 , 

 where p = 2a 2 + /3 2 . Hence p 1 = 2m 4 , p =F 1 = 4n 4 . Thus m 4 =F 1 = 2n\ which 

 is impossible unless m 2 = 1, whence x= 1 or 7. 



Genocchi 23 cited his 43 paper of Ch. XVI in which he proved that 

 2r 4 +2s 4 =t=D, whence p 4 +<r 4 =t=2D, so that Pepin's 21 condition ?7i 4 +l = 2(n 2 ) 2 

 requires m 2 = l. 



S. Realis 24 gave a discussion quite similar to that by Genocchi. 22 



E. Turriere 25 treated the system x = 2y*-l, x 2 = 2z 2 -!. 



A NUMBER AND ITS SQUARE BOTH SUMS OF TWO CONSECUTIVE SQUARES. 



E. de Jonquieres 26 gave a proof, valid only when y is a prime, that y 

 and y 2 are both sums of two consecutive squares only when y = 5. Like 

 errors invalidate his 27 result that if a number and its square are both 

 expressible in the form x 2 +t(x+l)~, then t = l, 2, 4, 5, 7, or 9. Cf. Lucas, 20 

 and papers 89, 90 of Ch. XVI. 



T. Pepin 28 reduced the problem to a certain quartic which he did not 

 solve completely. If y = P 2 -\-Pl, all decompositions of y 2 into a sum of 

 two relatively prime squares are given by y 2 = (P 2 Pi) 2 +(2PPi) 2 . Taking 

 2PPi and (P 2 P?) as consecutive integers, de Jonquieres assumed that 

 P and PI must be consecutive. While this condition is necessary if y 

 is a power of a prime or the double of such a power, it is in general not 

 necessary. 



E. Catalan 29 asked for numbers 2x expressible as a sum of squares of 

 two consecutive odd numbers, while (2#) 2 is a sum of squares of two con- 

 secutive even numbers, citing the case 2x = 10. G. C. Gerono 30 proved that 

 2x = 10 is the only solution of the equivalent system 



22 Nouv. Ann. Math., (3), 2, 1883, 306-10. 

 * 3 Bull. Bibl. Storia Sc. Mat., 16, 1883, 211-2. 



24 Ibid., p. 213. Reproduced, Sphinx-Oedipe, 4, 1909, 175-6. 



25 L'enseignement math., 19, 1917, 243^. 



28 Nouv. Ann. Math., (2), 17, 1878, 219-20, 241-7, 289-310; (2), 18, 1879, 464-5. Cf. Meyl 30 

 of Ch. IV. 



27 Ibid., (2), 17, 1878, 419-24, 433-46. Cf. Assoc. frang. av. sc., 7, 1S78, 40-49. 



28 Atti Accad. Pont. Nuovi Lincei, 32, 1878-9, 295-8. 



29 Nouv. Ann. Math., (2), 17, 1878, 518. 



30 Ibid., 521. 



