10 HISTORY OF THE THEORY OF NUMBERS. [CHAP. 1 



which for n = 2, 3, 4 include the polygonal numbers and the pyramidal 

 numbers of the first and second kind, the number of sides being a + 2. 



L. Euler 56 investigated polygonal numbers which are also squares. 

 The problem is a special case of that to make a quadratic function a square. 

 The triangular numbers equal to squares are those with sides 0, 1, 8, 49, 

 288, 1681, 9800, and equal the squares of 0, 1, 6, 35, 204, 1189, 6930, 

 The xih polygonal number with I sides is {(1-- 2)x 2 - (I - 4)z}/2. To 

 make it a square, set 2(1 - 2)p 2 + 1 = q 2 . Then the product of the 

 polygonal number by 4 is the square of 0, (I 4)p, 2(1 4)pg, if 





Euler gave a law for the derivation of any solution x in terms of two solu- 

 tions. It remains to make the expressions (4) integers. For Z = 5, q is 

 to be chosen from 1, 5, 49, and hence p from 0, 2, 20, -. The first 

 fraction (4) is here* (1 g)/6 and is an integer for q = 49, whence x = 8. 

 But Euler had previously stated that, for I > 4, q was to be taken negative. 

 The value q = 5 gives x = 1 and the pentagonal number 1. 



Euler 57 proved Fermat's theorems that no triangular number except 

 unity is a cube (since x 6 db y 6 is not a square), and no triangular number 

 x(x + l)/2 > 1 is a fourth power. According as x is even or odd, x/2 or 

 (x -f l)/2 must equal a fourth power m 4 , if the A is to be a fourth power. 

 Thus 2ra 4 1 = n 4 . But he had just proved that 2n 4 =b 2 = D only 

 when n = l, whence m = or 1, x = or 1. 



Abbe" Deidier 58 gave the simplest properties of polygonal numbers and 

 derived central polygonal numbers as follows: adding unity to the products 

 of the triangular numbers 0, 1, 3, 6, 10, by 3, 4 or 5, we get central tri- 

 angular, square or pentagonal numbers, respectively. 



We shall now quote from the correspondence 59 between Euler and Gold- 

 bach remarks on polygonal numbers, reserving for later use the comments 

 in which the interest is chiefly on sums of squares. June 25, 1730 (p. 31), 

 Euler noted that (x 2 + x)/2 equals (6/7) 4 for x = 32/49, but said this does 

 not disprove Fermat's assertion that no (integral) triangular number is a 

 biquadrate. Aug. 10, 1730 (p. 36), Euler noted that if 



a = (3 + 2V2)*, b = (3 -2A/2)", 



the square of (a 6)/(4 ^2) is a triangular number with the side (a+6 2)/4 

 [evident since db = 1]. Chr. Goldbach stated April 12, 1742 (p. 122) that 

 4mn - - m - - n a ^ A. Euler remarked May 8, 1742 (p. 123) that 

 4mn m n is not a heptagonal number. June 7, 1742 (p. 126), Gold- 



s' 1 Comm. Acad. Petrop., 6, 1732-3, 175; Comm. Arith. Coll., I, 9. Cf. Euler. 79 



* Thus q = I - 6x so that 6p 2 + 1 = g 2 becomes p 2 = - 2x + 6z 2 . Hence p = 2P, 



P 2 = (3z 2 x)/2, and we have returned to the problem from which we started. 

 "Comm. Acad. Petrop., 10, 1738, 125; Comm. Arith. Coll., I, 30, 34. Proof republished 



by E. Waring, Medit. Algebr., ed. 3, 1782, 373. 



68 Suite de Parithm6tique des g6ometres, Paris, 1739, 352-365. 



69 Correspondance Math6inatique et Physique (ed., P. H. Fuss), St. P6tersbourg, 1, 1843. 



