iv PREFACE. 



two centuries later by Roman and Hindu writers. The most important 

 theorem on the subject is that first stated by Fermat: Every positive integer 

 is either triangular or a sum of 2 or 3 triangular numbers; every positive 

 integer is either a square or a sum of 2, 3, or 4 squares ; either pentagonal 

 or a sum of 2, 3, 4, or 5 pentagonal numbers; and similarly for any polygonal 

 numbers. Throughout his half century of mathematical activity, the great 

 Euler was engaged on the subject of polygonal numbers and solved many 

 questions concerning them, but was able to prove Fermat's above theorem 

 only for the case of squares, and noted that the theorem for the case of 

 triangular numbers is equivalent to the fact that every positive integer of 

 the form 8n + 3 is a sum of three squares. This fact is a case of the 

 theorem that every positive integer, not of one of the forms Sn + 7 and 4n, 

 is a sum of three squares, which was proved in a complicated manner by 

 Legendre in 1798 and more clearly by Gauss in 1801, by means of the theory 

 of ternary quadratic forms. Gauss showed how to find the number of 

 ways in which a number N is a sum of three triangular numbers, by means 

 of the number of classes of binary quadratic forms of determinant 8N 3. 



Cauchy gave in 1813-15 the first proof of Fermat's theorem that every 

 number is a sum of m w-gonal numbers (all but four of which may be taken 

 to be or 1). Legendre immediately simplified this proof and showed that 

 every sufficiently large number is a sum of four or five m-gonal numbers 

 according as m is odd or even. In 1892 Pepin gave another proof of 

 Cauchy's result. In 1873 Realis proved that every positive integer is a 

 sum of four pentagonal or hexagonal numbers extended to negative argu- 

 ments. In 1895-96 Maillet proved that every integer exceeding a certain 

 function of the relatively prime odd integers and /3 is a sum of four numbers 

 of the form %(ax*-\-px}; also, if <j>(x) = aoX 5 -}- . . . +a 5 , where the a's are 

 given rational numbers, is integral and positive for every integer x suffi- 

 ciently large, then every integer exceeding a fixed function of the a's is a 

 sum of at most v positive numbers <f>(x) and a limited number of units, 

 where v = 6, 12, 96, or 192, according as the degree of <f> is 2, 3, 4, or 5. 



From formulas in his treatise on elliptic functions of 1828, Legendre 

 concluded that the number of ways in which N is a sum of four triangular 

 numbers equals the sum of the divisors of 2TV+1, and found the number of 

 ways in which TV is a sum of eight triangular numbers. In 1918 Ramanujan 

 obtained expressions for the number of representations of any number as a 

 sum of 2s triangular numbers. 



In 1772 J. A. Euler, the son of L. Euler, remarked that, to express every 

 number as a sum of squares of triangular numbers, at least twelve terms 

 are required, and stated that, to express every number as a sum of figurate 

 numbers 



1, n-\-a, 



1-2 1-2-3 



at least a-f 2n 2 terms are necessary. About the same time, N. Beguelin 

 stated erroneously that at most a+2n 2 terms are sufficient. In 1851 



