CHAP. I] POLYGONAL, PYRAMIDAL AND FIGURATE NUMBERS. 25 



S. Re"alis 129 proved that every integer is a sum of four numbers of the 

 form (3z 2 z)/2 and also of four of the form 2z 2 z, i. e., pentagons and 

 hexagons extended to negative arguments. Use was made of the theorems 

 that any odd number <o is a sum of four squares the algebraic sum of whose 

 roots is 1 or 3, and its double 2co is a sum of four squares the algebraic sum 

 of whose roots is zero. Further, every odd number divisible by h, or the 

 double of every odd number divisible by the even number h, is a sum of 

 four polygonal numbers of order h 4- 2, extended to negative arguments. 



E. Lucas 130 stated that [cf. Leybourn 80 ] I 2 + + n 2 is a square only 

 when n = 24 [and n = 1], and is never a cube or fifth power. A triangular 

 number [> 1] is never a cube, biquadrate or fifth power [Euler 57 ]. No 

 pyramidal number is a cube or fifth power, or a square with the exception of 



2-3-4 48-49-50 _ 



1-2-3" 1-2-3 



Hence except for these and for the pile 24-25-49/6 = 70 2 with a square 

 base, no pile of bullets with a triangular or square base contains a number 

 of bullets equal to a square, cube or fifth power. 



Lucas 131 stated and proved incompletely that the [pyramidal] number 

 z(a;+l)(2z+l)/6 of bullets in a pile, whose base is a square with z to a side, 

 is a square only when x = 1 or 24 (see papers 130, 132, 137-8). 



T. Pepin 132 noted that one case of Lucas' proof of the last result leads 

 to an equation 9r 4 12/V 2 4f 4 = R 2 , not treated by Lucas when / and 

 R are divisible by 3. Pepin found an infinitude of solutions in this case. 

 G. N. Watson 1320 noted the solution r = 5, / = 3, R = 51, and 1326 proved 

 Lucas' 131 theorem by use of elliptic functions. 



Lucas 133 stated that the number of bullets in a pile with a square or 

 triangular base is never a cube or fifth power. Moret-Blanc 134 gave a proof. 



Moret-Blanc 135 noted that the tetrahedral number n(n + l)(n + 2)/6 

 is a square for n = 1, 2, 48. Lucas stated that it is a square only then, 

 a fact proved by A. Meyl. 136 



E. Fauquembergue 137 and N. Alliston 138 proved that I 2 + + n 2 =(= D 

 if n > 24. Cf. Lucas 131 and the papers cited on p. 26. 



129 Nouv. Ann. Math., (2), 12, 1873, 212; Nouv. Corresp. Math., 4, 1878, 27-30. 



130 Recherches sur 1'analyse indeterminSe, Moulins, 1873, 90; extracted from Bulletin de la 



societe d'emulation Dept. de 1'Allier, Sc. Bell. Let., 12, 1873, 530. 



131 Nouv. Ann. Math., (2), 14, 1875, 240; (2), 16, 1877, 429-432. The proof by Moret- 



Blanc, (2), 15, 1876, 46-8, is incomplete (as noted p. 528). 



132 Atti Accad. Pont. Nuovi Lincei, 32, 1878-9, 292-8. 



i32a p roc London Math. Soc., Record of Meeting, March 14, 1918. 

 132!> Messenger of Math., 48, 1918, 1-22. 



133 Nouv. Ann. Math., (2), 15, 1876, 144 (Nouv. Corresp. Math., 2, 1876, 64; 3, 1877, 247-8, 



433, and p. 166 for incomplete proof by H. Brocard). 

 Ibid., (2), 20, 1881, 330-2. 

 Ibid., (2), 15, 1876, 46. 



136 Ibid., (2), 17, 1878, 464-7. 



137 L'intermediaire des math., 4, 1897, 71. 



138 Math. Quest. Educ. Times, 29, 1916, 82-3 (for n < 10 21 by J. M. Child, 26, 1914, 72-3; 



for n < 10 12 by G. Heppel, 34, 1881, 106-7). 



