CHAP. I] POLYGONAL, PYRAMIDAL AND FIGURATE NUMBERS. 35 



He treated (pp. 97-101) the decomposition of various types of numbers 

 into a sum of three triangular numbers. 



The ordinary definitions of polygonal and figurate numbers were repeated 

 by L. Tenca 221 and E. A. Engler. 222 



L. Tits 223 solved Emmerich's 189 equation for y, made the radical rational 

 and was led in both cases to t z Sv 2 = 1. 



E. Barbette 224 gave many numerical examples in which a sum of n-gonal 

 numbers equals an n-gonal number. 



L. von Schrutka 225 found that |{(|T) 2 + U 2 } is not expressible in a 

 single way as a sum of two numbers of the form T(x z + x)/2 + Ux unless 

 T{2 = 3 or 5. In the first case it is shown that, if p is a prime = 5 (mod 

 12), (p 2)/3 can be expressed in one and but one way as a sum of two 

 8-gonal numbers 3z 2 2x. He gave an analogous theorem for 12-gonal 

 numbers 5x 2 4x, and one for numbers 5z 2 2x. 



A. Ge*rardin 226 solved n 2 + 2 e n = A* for x by setting n = xpfq. He 

 (p. 128) reduced A* A y = A z2+ to 2 Ax + 1 = A and noted the solutions 

 A x = 10, 45, A v = 21, 91. 



L. Bastien 227 noted that x 4 - y* = A 2 if z = x 2 + y z and x* - 3y* = 1, 

 or if z = (x 2 + ?/ 2 )/X, z -[-I = 2\(x 2 y*) or vice versa, whence 



(2X 2 - l)z 2 - (2X 2 + l)y 2 = dz X. 



G. Me'trod 228 noted that A M - A. = x 5 if (u - v)(u + v + 1) = 2z 3 , 

 whence 2x* is to be expressed as a product of two distinct factors, one even 

 and one odd. 



F. Mariares 229 noted that the sum of 1,2, - - -, n is n(n + l)/2 since the 

 sum duplicated makes a rectangle of n by n + 1. Again, 



1 + 3+6+ ... + ^ + 1 ) ==22 + 42+ ... 

 or 



according as n is even or odd. Hence 



Ai + A 2 + + An-i + Ai + + An = 



ft=l 



Numbers simultaneously triangular and pentagonal have been treated. 23 



221 II Boll, di Mat. Sc. Fis. Nat., 12, 1910-11, No. 1, p. 16, No. 3, p. 24. 



222 Trans. St. Louis Acad. Sc., 20, 1911, 37-57. 



223 Mathesis, (4), 1, 1911, 74-5. 



224 L'enseignement math., 14, 1912, 19-30. Cf. Barbette. 217 

 226 Monatshefte Math. Phys., 23, 1912, 267-273. 



226 Sphinx-Oedipe, 8, 1913, 110, 121-2 (1907-8, 173; 1911, 75). 



227 Ibid., 156, 172-3. 



228 Ibid., 174. 



229 Revista Soc. Mat. Espanola, 2, 1913, 333-5. 



Mathesis, (4), 3, 1913, 20-22, 80-81. a. Euler. 70 



