CHAP. I] POLYGONAL, PYRAMIDAL AND FIGURATE NUMBERS. 17 



To find triangular numbers whose triples are triangular, Euler proved 

 that 3(z 2 + x) = y 2 + y has only the solutions 



y = ~- -\, s = (3 Vs - 5) (2 - V3), 



for n = 0, 1, d= 2, . Examples are # = 1, y = 2; # = 5, y = 9. 



It was proposed as a prize problem in the Ladies' Diary for 1792 

 to find n (n > 1) such that I 2 + 2 2 + + n 2 = D. The sum is 

 n(n + l)(2n + l)/6. T. Leybourn 80 took 2n + 1 = z 2 , whence (z 4 - l)/24 

 is to be a square ?/ 2 . Thus 2 4 = 24?/ 2 + 1 = D = (xy I) 2 , say. Thus 

 y = 2x/(x 2 24) > 0. It is stated that x = 5 or 6. Since x = 6 is 

 excluded, n = 24. C. Brady took n = 6r 2 . Then the condition is 

 (6r 2 + l)(12r 2 + 1) = D. Thus (9r 2 + I) 2 - (3r 2 ) 2 = D, so that 9r 2 + 1 

 and 3r 2 equal the hypotenuse and one leg of a right triangle. Thus the 

 other leg is 9r 2 1, whence r = 2, n = 24. 



A. M. Legendre 81 proved Fermat's theorems that no triangular number 

 x(x + l)/2, except unity, is a fourth power or cube. For, in the first 

 problem, x or x + 1 is of the form 2m 4 , whence either 1 = n 4 2m 4 , 

 contrary to 1 + 2m 4 =}= D, or 1 = 2m 4 n 4 , m 8 n* = (m 4 I) 2 , con- 

 trary to p 4 ft 4 =J= D unless m = I = x. In the second problem, one of 

 1 + x, x is a cube and the other the double of a cube, whence n 3 1 = 2m 3 , 

 which is impossible if n 4= 1. 



C. F. Gauss 82 proved by means of the theory of ternary quadratic forms 

 that every number n = 8M + 3 is a sum of three odd squares, so that, by 

 (5), M is a, sum of three triangular numbers. The number of ways M 

 can be so decomposed depends in a definite manner on the prime factors 

 of n and the number of classes of binary quadratic forms of determinant n. 



G. S. Kliigel 83 gave an account of figurate, polygonal, polyhedral, and 

 pyramidal numbers P r m of the first order, those of the second order being 



P l m + '+ P r m , etc. 



John Gough 84 attempted to prove Fermat's theorem that every number 

 is a sum of m m-gonal numbers. P. Barlow 85 noted that the first three 

 propositions by Gough are correct, but are not used in his defective proof 

 of Fermat's theorem, while various points are not proved, as the Cor. 2 

 to Prop. 4: every number is a sum of a limited number of polygonal num- 

 bers. As to Gough's reply (pp. 241-5), Barlow 86 stated that the defense 



80 Ladies' Diary, 1793, p. 45, Quest. 953. T. Leybourn's Math. Quest, proposed in the 



Ladies' Diary, 3, 1817, 256-7. Cf. Lucas, papers 130-8. 



81 Theorie des nombres, 1798, 406, 409; ed. 2, 1808, 345, 348; ed. 3, II, 1830, arts. 329, 335; 



pp. 7, 11. German transl. by Maser, 1893, II, 8, 13. 



82 Disquis. Arith., 1801, art. 293; Werke, I, 1863, 348; German transl. by Maser, 1889, p. 334. 



83 Math. Worterbuch, 2, 1805, 245-253; 3, 1808, 825-8, 931. 



84 Jour. Nat. Phil., Chem., Arts (ed., Nicholson), 20, 1808, 161. 



85 Ibid., 21, 1808, 118-121. 



86 Ibid., 22, 1809, 33-35. 

 3 



