34 HlSTOEY OF THE THEORY OF NUMBERS. [CHAP. I 



It is stated 211 that every nth power is a sum of n A's =1= 0; for example, 



3 4 = 55 + 15 + 10 + 1, 



5 5 = 2850 + 210 + 45 + 2-10 = 3003 + 105 + 10 + 6 + 1. 

 G. Nicolosi 212 gave an elementary proof of Cantor's result that 



%(x + y)(x + y + l)+y = a 



has one and but one set of integral solutions. Solving for y we see that 

 Sx + Sa + 9 must be a square v?. Thus x is integral only if u z 1 = 86, 

 whence = t(t + l)/2. 



C. Burali-Forti 213 noted relations like 



p n m - pi = (m - r)pr l , P n m + P r m = P n m + " ~ nr(m - 2), 

 mp n m np = \mn(m n). 



A. Cunningham 214 gave a method of expressing an integer as a sum of 

 three triangular numbers. 



P. Bachmann 215 gave an introduction to polygonal and figurate numbers. 



T. Hayashi 216 proved that the quadruple of a number a(a + ft) (a + 20) /6 

 and hence of a pyramidal number is not a cube, by making use of the known 

 impossibility of x 3 + y z = 3z 3 . 



E. Barbette 217 summed the pth powers of consecutive n-gonal numbers, 

 found sums of pth powers of n-gonal numbers equal to a pth power of an 

 n-gonal number, found cases with n ^ 6 in which a sum of two n-gonal 

 numbers is n-gonal, and gave a table of the first 5000 triangular numbers. 



H. Brocard 218 solved 10 A z + A y = z 2 for x and made the radical 

 rational. 



L. Aubry 219 noted that A x -iA x A x +i = D if (x - 1)0 + 2) = 2*/ 2 , 

 whence u? Sv z = 1, where 2x + 1 = 3u, y = 3v. The solutions are 

 known to be u 1, 3, 17, , u n = 6w n -i w B -2. 



A. Ge"rardin 220 collected recent problems on triangular and pentagonal 

 numbers. He noted (p. 70) that 



3 2n z = A a - A fc> a = 3 n x + (3" - l)/2, b = Z n x - (3" + l)/2. 

 Let a, b become c, d when x = y z ; then 



A(c) + A(d) = A(d - 3"y) + A(d + 3"y). 



211 Sphinx-Oedipe, 1906-7, 31, 46. 



212 II Pitagora, Palermo, 15, 1908-9, 15-17. In Suppl. al Periodico di Mat., 1908, fasc. 5-6, 



there is a proof by triangular numbers. 



213 Ibid., 16, 1909-10, 135-6. 



214 Math. Quest. Educ. Times, (2), 15, 1909, 44-5. 

 216 Niedere Zahlentheorie, 2, 1910, 1-14. 



216 Nouv. Ann. Math., (4), 10, 1910, 83. 



217 Les sommes de p-iemes puissances distinctes e'gales a une p-i&me puissance, Liege, 1910, 



154 pp. Extract by Barbette. 224 



218 Sphinx-Oedipe, 6, 1911, 29-30. 



219 Ibid., 187-8. Problem of Lionnet, Nouv. Ann. Math., (3), 2, 1883, 310. 



220 Sphinx-Oedipe, 1911, 40-3, 57-8, 81-6, 113-21, 129-32. 



