CHAP. IX] RELATIONS BETWEEN SQUARES. 321 



Catalan 83 proved the last result and (p. 106) gave a long identity 

 furnishing particular solutions of u 2 = x\ + + xl- If an odd number 

 N is a ED and if n is the number of equal or distinct prime factors of N, then 

 N 2 is a sum of k squares =J= 0, k = 2, 3, , n + 1. 



R. W. D. Christie 84 noted equal sums of four or more squares. 



A. Martin 85 noted that 2 2 + 3 2 + 6 2 = 7 2 , I 2 + 2 2 + 4 2 + 6 2 + 8 2 = II 2 , 



I 2 + 2 2 + + 50 2 - 206 2 = 1 + 2 2 + 22 2 = 5 2 + 8 2 -f 20 2 . 



He 88 stated that one can find several sets of 50 squares whose sum is 23 1 2 , 

 that I 2 + 2 2 + + 24 2 = 70 2 , and similar results. Cf. Lucas. 70 

 F. Tano's method to find an infinitude of solutions of 



= a, 



when k is of the form (3 n - l)/2, is given in Ch. XII. 207 



A. Martin 87 found many sets of squares whose sum is a square by 

 means of the methods of Aida 59 and Gray, 80 and by seeking to express 

 S n 6 2 as a sum of distinct squares ^ n 2 , where b 2 lies between n 2 and 

 S n = I 2 + + n 2 . He noted that the sum of n consecutive squares is 

 not a square for 2 < n < 11, and gave solutions for n = 11, 23, 24, 26, 

 etc. [cf. Lucas 70 ]. He gave solutions of 



s n - x 2 = n, Sn + 1 = n, s n - s m - x 2 = n, 



and tabulated the values of S n for n < 400. 



E. Catalan 88 noted that, if N db 1 are primes and N 4= 2, 2N 2 + 2 

 is a sum of 2, 3, 4, and 5 squares. 



E. Fauquembergue 89 and others noted the identities 



++ ay = (al + - + a} - a? +1 ----- <# + Z E 



r=l =+! 



(!+ + a*) 2 = W + al + oj - al - al) 2 + 4(0^4 a 3 a 5 ) 2 



=F 



P. H. Philbrick 90 noted that we may find n squares whose sum is a 

 square by Aida's 59 method or by starting with a sum S of n 1 squares 

 such that S is a product of two factors a and Z>, both even or both odd, and 

 applying 



R. J. Adcock 91 noted that, tfs = x + y-}-z + v, 

 x 2 s 2 + y 2 s 2 + 2 2 s 2 + y 2 s 2 4- (xy + xz + xv + yz + yv + zv) 2 = 



8S Atti Accad. Pont. Nuovi Lincei, 37, 1883-4, 53. 



M Math. Quest. Educ. Times, 49, 1888, 159-173; French transl., Sphinx-Oedipe, 7, 1912, 



177-87 



85 Bull. Phil. Soc. Wash., 10, 1888, 107 (Smithsonian Miscel. Coll., 33, 1888). 

 **Ibid., 11, 1892,580-1. 



87 Math. Mag., 2, 1891-3, 69-76, 89-96, 137-140. 



88 Mathesis, (2), 3, 1893, 235. 



89 Mathesis, (2) 4, 1894, 277; 6, 1896, 101. 



90 Amer. Math. Monthly, 1, 1894, 256-8. 



91 Ibid,, 2, 1895, 285. 

 22 



