CHAP. I] POLYGONAL, PYRAMIDAL AND FIGURATE NUMBERS. 37 



bergue, 4, 1897, 159-162, was corrected later, 5, 1898, 257. P. F. Teilhet, 11, 

 1904, 11-12, verified that aside from 0, 1, 6 there is no A with fewer than 

 660 digits whose square is a A. 



G. de Rocquigny, 2, 1895, 394, noted that every triangular number 

 except 1 and 6 is a sum of three, each =f= 0, since 237 



A(3p - 1) = 2A(2p - 1) + A(p), A(3p) = 2A(2p) + A(p - 1), 



A(3p + 1) = A(2p) + A(2p + 1) + A(p). 



On A's expressed as a sum of two or three A's, see 4, 1897, 158. For solu- 

 tions of (x + I) 3 - x 3 = A y , see 4, 1897, 262-4; 5, 1898, 18, 110-1 (and 

 Mathesis, (2), 8, 1898, 126). 



E. Fauquembergue, 4, 1897, 209, noted that A x + A y = z 2 is equivalent 

 to (2x + I) 2 + (2y + I) 2 = (2z + I) 2 + (2z - I) 2 , which by Euler's for- 

 mula for the product of two sums of two squares has the solution 

 2x + 1 = ac + bd, 2y + 1 = be - ad, if be + ad = ac - bd + 2. Cf . 

 Ge'rardin 12 of Ch. XXIV. A. Palmstrom, 210, noted that the problem is 

 equivalent to 



(x + y}(x - y + 1) = 2(0 + y)(z - y}. 



On A* + A y = z 3 see 7, 1900, 250. E. B. Escott, 11, 1904, 82, noted 

 that A x + A y = z 5 is equivalent to (2x + I) 2 + (2y + I) 2 = 2(4z 5 + 1), 

 a necessary and sufficient condition for which is that every prime factor 

 of 4z 5 + 1 be of the form 4n + 1 ; and gave solutions for z = 1, 4, 6, 9, 12, 16. 



On A x = y 2 + z 2 see 3, 1896, 248; 4, 1897, 129-132, 255. 



Any number N is a sum of three pentagons (3x 2 =h x)/2 since 



3 = Z (6z d= I) 2 



is solvable, 4, 1897, 157. On A' + A" = A, 4, 1897, 158; 5, 1898, 70. 

 The sum n(n + l)(w + 2)/6 of the first n A's is a A for n = 1, 3, 8, 20, 34, 

 but for no further n < 316, 4, 1897, 159; 6, 1899, 176; 7, 1900, 192; 16, 

 1909, 236; 17, 1910, 110; and is a square for n = 1, 2, 48, but for no others 

 < 10 12 , 9, 1902, 279; 10, 1903, 235. The sum n(n + l)(2n + l)/6 of the 

 first n squares is a A for n = 1, 5, 6, 85 by 6, 1899, 175; 7, 1900, 211; 9, 

 1902, 278. 



P. Tannery, 5, 1898, 280, and C. Berdelle, 7, 1900, 279, gave algebraic 

 and geometric proofs that, aside from 6, every p-gonal number is a sum of 

 p 2 triangular numbers > 0. 



A prime Qn + 1 = 3p 2 + g 2 is a sum of 3 A's > 0, 4, 1897, 119. Since a 

 prime 8n + 1 equals 8m 2 + (2p + I) 2 , it equals Aim + D + P, where 

 P = m(6m - 1) is pentagonal, 8, 1901, 183. 



G. de Rocquigny, 7, 1900, 65, 195; 8, 1901, 52; 9, 1902, 116, 176, 230; 

 10, 1903, 5-6, 40, 122, 205-6, 285, 300-2; 11, 1904, 99, 150, 158, 163-4, 

 189, 214, 237; 15, 1908, 181, stated many theorems of the following type: 

 every sixth power is a sum of a square, cube and triangular (or hexagonal) 

 number; every number > 7 is a sum of three A's and three squares each 

 =1= 0. A. Gerardin, 18, 1911, 177-184, 199, 275, discussed these theorems. 



237 Same by R. W. D. Christie, Math. Quest. Educ. Times, 69, 1898, 48. 



