In 1921 Kampke considered any polynomial P(x) with rational coefficients whose values are integers Z o for every integer x 0, and proved that every integer po is a sum of s such values of P(x) and e numbers o or 1, where s and e are finite integers depending on P(x), but not on p. This existence proof gives no clue to the values of s and e. He also generalized Cauchy's lemma from 2 to n simultaneous equations where • • • , /„ lie between specified limits, and N is to be chosen.
The complete solution for quadratic functions was given by Dickson in 1928. Consider f(x) = where m, c are integers and m> o, t> o, A quadratic function takes integral values > o for every integer x Z o if and only if it is either f(x) or f(x- k) with k Z m>_t. Since c is the least such value of f(x-k), sc is the least sum of s values. The problem is to find the least integer such that every integer _.sc is a sum of s values of f(x-k) for integers and numbers o or 1. We may take c= o without loss of generality. First, let 5. If k=o, then is t or m - (s - 3)4 according as m< (s- 2)t or not. If k>o and m > 21, then ist or m- (s+ 1) t, according as m< (s+2)t or not. When m <2t, write q for m-t. If k > 2, E, is q or 1-3q, according as (s + r)q or not. When k=1,
if t 3q, if t 3q. But if k=i, 6, or t-(s-2)q, according as t__(s--1)q or not. We now know the minimum L for all s Z 5 of the number s+e, of summands. Whenever 4+e4 is less than L, it is exactly L-i. The minimum of all for s I is L or L- i except when k = o, m and then is L- 2.
Every integer p o has been recently proved to be a sum of ten pyramidal numbers for integers x o also a sum of nine if p is sufficiently large.
If in the formula for polygonal numbers we employ negative as well as positive integral values of x, we obtain the generalized polygonal numbers. Every positive integer is a sum of N generalized polygonal numbers of order k, where N=3 for k = 3, Sor6;N=4fork=4, 7 or 8;N=k-4 for k>8.
Consider the positive binary quadratic forms q and the in tegers s such that every positive integer p is a sum of s values of q, The case p =1 requires that q shall represent 1. Hence we may take a= o or 1. Then each p is a sum of four values of q with y = o. Thus the cases s= 2 and s =3 alone are interesting. Whether a is o or 1, every p is a sum of two values of q if and only if b= I, 2, or 3, and a sum of three values of q if and only if b=i, • • • , 7.