36 HISTORY OF THE THEORY OF NUMBERS. [CHAP. I 



* S. Minetola 231 gave a combinatory definition of the numbers in 

 TartaghVs triangle. 



N. Alliston and J. M. Child 232 proved that no triangular number > 1 

 is a biquadrate. 



An anonymous writer 233 proved that if a number 4n + 1 is a C3 , it is a 

 sum of two triangular numbers c(c + l)/2 and d(d + l)/2, where d may be 

 negative; then c and d are of the same parity. 



G. Metrod 234 stated that, if p n q is the nth polygonal number of q sides, 

 the g.c.d. of p* and p n q +l is the g.c.d. of n + 1 and q 3 unless the latter 

 are even and then is the g.c.d. of (n + l)/2 and q - 3. The g.c.d. of p n q 

 and p" + i is n or n/2 according as n is even or odd. 



A. Gerardin 2340 noted that 2 A* 1 is a prime for x ^ 9. He 2346 gave a 

 series for A* A = A* with the law of recurrence z n +i = 6z n z n -\ + 2, 

 2 = 3^ jgj = 20. He 234c gave a general solution of A + A& = e 2 + 2 Ad, a 

 special case having been noted by Euler 59 , and noted the examples a=2s+l, 

 6 = 4s, d = 3s, e = s + 1 ; a = 6s + 2, 6 = 4s - 1, d = 5s + 1, e = s - 1. 



E. Bahier 235 found sets of three m-gonal numbers in arithmetical pro- 

 gression: pn + p v m = 2p? n . Multiply each p by S(m - 2) and add (m - 4) 2 

 to each product. By Diophantus' relation (1), we get 



Pi + PI = 2Pl, P^(m- 2)(2X - 1) + 2. 

 Hence, by Ch. XIV, 



P, = (x 2 - 2xy - ?/ 2 ), P M = z 2 + 2/ 2 , P, = x 2 + 2xy - y\ 



Then X, ju> ^ are found in terms of x, y, m by use of the above equation 

 defining P A . The conditions that X, /x, v be positive integers are discussed 

 at length. 



S. Ramanujan 2350 obtained expressions for the number of representations 

 of n as a sum of 2s triangular numbers. 



NOTES* FROM L'INTERMEDIAIRE DES MATHEMATICIENS. 



A. Boutin, 236 1, 1894, 91; 2, 1895, 31, noted that the square of each term 

 of the series 0, 1, 6, 35, -,u n = u n -\ u n -z, is a triangular number 

 A, and stated that the A's in this series (viz., 0, 1, 6 up to u^) are the only 

 A's whose square is a A. He gave all solutions x = 8, 800, of 

 x z + Ax = D and stated that y* d= 1 = A 2 only for y = 1, 3, 16, 20; 

 x = 0, 1, 7, 90, 126. An incorrect solution of the latter by E. Fauquem- 



231 Boll, di Matematica, Roma, 12, 1913, 214-22. 



232 Math. Quest. Educ. Times, 25, 1914, 83-5. 



233 Nouv. Ann. Math., (4), 14, 1914, 16-18. 



234 Sphinx-Oedipe, 9, 1914, 5. 

 2340 Ibid., p. 41. 



2346 Ibid., p. 75, p. 146. 

 234c Ibid., p. 129. 



236 Recherche . . . Triangles Rectangles en Nombrcs Entiers, 1916, 217-233. 

 2360 Trans. Cambridge Phil. Soc., 22, 1918, 269-272. 



*For a more extended account see Ge"rardin. 220 The present notes were obtained inde- 

 pendently. 

 236 Jour, de math. 616m., (4), 4, 1895, 222. Cf. Lionnet. 168 



