CHAP. I] POLYGONAL, PYRAMIDAL AND FIGURATE NUMBERS. 33 



E. Grigorief 198 discussed Fermat's theorem that every number is a 

 sum of three A's. 



L. Kronecker 199 gave a brief history of polygonal numbers. 



J. J. Barniville 200 evaluated series involving figurate numbers, such as 

 I 3 + (I 3 + 3 3 )2~ 1 + (I 3 + 3 3 + 6 3 )2~ 2 + (I 3 + 3 3 + 6 3 + 10 3 )2~ 3 + =6416. 



A. Cunningham 201 noted that A* + A! = 2 A* if Ax = A 2 , A v = 7?A 2 , 

 where (, 77) = (1, 1), (7, 5), (41, 29), (239, 169), etc. 



Cunningham and Christie 202 solved /xA* = vA y , which is equivalent to 

 (2x + I) 2 - v(2y + I) 2 = /* - v , by use of a solution of 2 - /i*y = 1. 



R. W. D. Christie 203 argued that no A is a cube > 1. 



A. Cunningham 204 noted that A a A* = AaAj, is equivalent to 



A a (X 2 - 1) = A a (F 2 - 1), 



whose solutions follow from the least solution of 2 A a A ^ 2 = 1. 

 Christie 205 noted that N = A 2a + A 2 & + A 2c implies 



2N + 1 = (a + b + c + I) 2 + (a - b - c) 2 + (a + b - c) 2 + (a - b + c) 2 , 



and similar formulas in which some of 2a, 26, 2c are replaced by odd numbers. 

 Cunningham 206 noted that, if x = f(10 B - 1), A* = 2- -21- -1 (n 

 two's and n one's). 



E. B. Escott 207 proved that 55, 66 and 666 are the only triangular 

 numbers, with fewer than 30 digits, consisting of a single repeated digit. 



F. Hroma"dko 208 noted that if At, , A w are any consecutive A's, 



AH - A? = (A 2 - AO 3 + (A. - A 2 ) 3 + + ( An ~ A-i) 3 . 

 L. von Schrutka 209 proved that, if I = p a m (mod jfc), then 

 m \ /m-4\ 2 \m \ m \ I 2 



and conversely if m/2 1 is prime to k, so that k is called regular. The 

 question of polygonal residues thus reduces to that of quadratic residues. 

 Irregular moduli k are treated on pp. 190-3. 



J. Blaikie 210 noted that \n(n + 1) is also a pentagonal number 

 ^m(3m 1) if 3y 2 x 2 = 2, where x = 6m 1, y = 2n + 1. From solu- 

 tions of the Pell equation p 2 3q 2 = 1, we get solutions x = I23q 

 y = 41p 71g of the former. 



198 Kazan Izv. fiz. mat. obsc. (= Bull. Math. Phys. Soc. Kasan), 11, 1901, No. 2, 64-69 



(in Russian). 



199 Vorlesungen liber Zahlentheorie, 1901, 17-22. 



200 Math. Quest. Educ. Times, 74, 1901, 80. 



201 Ibid., 65-6. 



202 Ibid., 87-8. 



203 Ibid., 75, 1901, 36. 



204 Ibid., 120-1. 



205 Ibid., (2), 1, 1902, 94-5; 6, 1904, 85-6. 



206 Ibid., 8, 1905, 25. 



207 Ibid., 33-4. 



208 Zeitschr. Math. Naturw. Unterricht, 35, 1904, 306. 



209 Monatshefte Math. Phys., 16, 1905, 167-193. 



210 Math. Quest. Educ. Times, (2), 9, 1906, 40-41. 

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