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



G. W. Kraft 61 and A. G. Kastner 62 proved that 



2 4m+l _ y.m _ I 



= A, 



9 2 



since (2 Zm l)/3 is an integer N. 



M. Gallimard 63 obtained "central polygons" by multiplying each term of 

 0, 1, 3, 6, 10, 15, by the number n of angles of any polygon whatever 

 and adding unity to each product. Given a central polygon, he treated 

 the problem to find the number of angles if the side be given, or vice versa. 



L. Euler 64 proved that a number not the sum of a square and a triangular 

 number A is composite; one not A + 2 A' is composite. 



Nicolas Engelhard 65 treated Plutarch's 6 questions on triangular numbers. 



Elie de Joncourt 66 gave a table of triangular numbers N(N + l)/2, N 

 up to 20000, and showed how the table may be used to test if a number 

 less than a hundred million is a square or not, and to extract square roots 

 approximately. 



L. Euler 67 noted that, if N - ab = A P + A g -f A r and p - q = a - b, 

 then N = A P+b + A P _ a + A r . N. Fuss, I, (pp. 191-6) also gave an 

 incomplete argument to show that N is a sum of three triangular numbers 

 if every integer < N is. Let N p = A a + A& + A c and 



p = (b a)n + n 2 



[a restriction]; then N = Ao-n + Ab+n + A c . He gave a similar in- 

 complete discussion of the problem to express N as a sum of m w-gonal 

 numbers, given that every integer < N is such a sum. He noted (p. 201) 

 that 9n + 5, 8; 49n + 5, 19, 26, 33, 40, 47; Sin + 47, 74; etc., are not 

 sums of two triangular numbers; thus, 49n + 19=A a +A& would imply 

 (2a + I) 2 + (26 + I) 2 = 8(49n + 19) + 2, whereas the factor 7 of the 

 latter is not a divisor of a sum of two squares. L. Euler (p. 214) noted that 

 A x A = A z is satisfied* if px(y + 1) = 2qz, qy(x + 1) = p(z + 1) ; the result- 

 ing values of z are equal if {(2g 2 p^x + 2q z }y = p 2 x + 2pq. L. Euler 

 (pp. 264-5, about 1775) noted that 



9A a + 1 = A3a+l, 49 A a + 6 = A 7 a+3, 



25 A a + 3 = A 5 a+2, 81 A + 10 = A 9 a+4. 



J. A. Euler 68 (the son of L. Euler) stated that to express every number 

 as a sum of terms of I 2 , 3 2 , 6 2 , 10 2 , 15 2 , , at least 12 terms are required. 



61 Novi Comm. Acad. Petrop., 3, ad annum 1750 et 1751, 112. 



62 Comm. Soc. Sc. Gottingensis, 1, 1751, 198. Cf. T. Pepin, Atti Accad. Nuovi Lincei, 32, 



1878-9, 298. 



" L'Algebre ou la Science du Calcul litteral, Paris, 2, 1751, 131-143. 

 M Novi Comm. Acad. Petrop., 6, 1756-7 [1754], 185; Comm. Arith. Coll., I, 192. 

 6B Verhandel. Hollandse Maatachappy Weetenschappe te Harlem, 3 Deel, 1757, 223-230; 



4 Deel, 1758, 21 (correction to p. 224). 



66 De Natura et Praeclaro Usu Simplicissimae Speciei Numerorum Trigonalium, Hagae 



Comitum, 1762, 267 pp. 



67 Opera postuma, 1, 1862, 190 (about 1767). 



* The least solution is x = 2, y = 5, z = 9, Sphinx-Oedipe, 1913, 90; 1914, 145. 

 88 Ibid., pp. 203-4, about 1772. 



