Classes of Functions Representation of Integers by Functions

We saw that all positive ternary forms of determinant 3 are equivalent to P or Q. To show that P represents every positive integer a not divisible by 3, Dirichlet used (I) with s= 1, t= o and b— 5 a multiple of 8. It is easily verified that Q— 5 is never divisible by 8. Since (I) is of determinant i and represents b, while Q does not represent b, (I) must be equivalent to P. But he did not prove that P represents every positive integer 9n+3.

Its proof requires a serious modification of his method and was given, along with a complete study of various new ternary forms, in several papers published in America in 1927. The complete theorem for P is that a positive integer can be rep resented by P if and only if it is not of the form The forms P and are called regular since all the positive integers not represented by one of them coincide with all the positive integers in certain arithmetical progressions. The form is regular with respect to even integers since it represents no number but represents all the remaining even integers. However, g is irregular with respect to odd integers. It was proved in 1927 that if k is any one of the odd integers 3, 7, 21, 31, 33, 43, 67, 79, 87,133, 217, • • • not repre sented by g, then every arithmetical progression which contains k will contain integers represented by g. Only a few forms are regular.

Positive Quaternary Quadratic Forms.-One of the most remarkable theorems is that every positive integer p is a sum of four squares. Diophantus seems to have recognized this fact. In 1659 Fermat stated that he possessed a proof by descent from p to smaller numbers, and a short proof of this kind was published in 1924. Euler tried for 4o years to find a proof. The first proof published was that by Lagrange in 1772. The next year Euler gave a much simpler proof which is still quoted in text-books.

Every

p is represented by each of (s=1, • • -,7). Except when p=4k(871+7), we may take u=o and apply the earlier result that p is a sum of three squares. We shall next prove that 8n+ 7 is represented, whence • • • , 2ku give a representation of (8n + 7). We have only to exhibit a value of u for which is positive and not of the form 7) and hence is a sum of three squares. For s= I, 2, 4, 5, or 6, take u = I. For s = 3, take u= I or 2 according as n = o or n>o. For s= 7, take u= I if n=o, 1, or 2; but take u= 2 if 3.

The same proof applies to

A similar proof shows that every p is represented by each of Besides these 54 forms there is no new form of the type which represents every p when a, b, c, d are positive integers. This is easily verified by using the values I, 2, 3, 5, 7, 10, 14, 15 of p.

Recently the problem has been completely solved for forms

q involving also products like xy. The main point in the solution is a simple application of the principles which have been de veloped above in detail. It will be explained for the important typical case q= T where T is a positive ternary quadratic form. Let m be the least positive integer represented by T. Unless m is I or 2, q would not represent 2. Let m= I and let d denote the determinant of 2T. The main theorem quoted under ternary forms shows that 2r (whose minimum is a= 2) is equivalent to a form 2T =F such that is the reduced binary form fin (2), where X = 2x-Fty±sz,t= o or 1, s = o or 1. By hypothesis, represents all positive integers. Hence rep resents all positive multiples of 4. The positive minimum of the reduced f is L. If L> 12, Q= 12 requires that f=o. But 12 = is impossible in integers. Hence L:5. 12. Since the coefficients of in and 4T are equal, P--FL is a multiple of 4. Hence the only possible values of L are 4, 8, 3, 7, II. In these respective cases the least positive multiple of 4 not represented by (or Q with z= o) is found to be 1=28, 56, 24, 84, 88, respectively. Since Q shall represent 1, we have f <1 for certain integers y and z such that zX o. Then (3) evidently implies that 2d Ll. Hence there is a limited number of values of d, and therefore of reduced forms f of discriminant -8d. This proves that the number of forms T is finite.

In addition to the above 54 forms involving only squares, there are exactly 299 further forms T which represent all positive integers. If we multiply them by 4, 8, 12, 28, or 44, we may com plete the squares and obtain forms involving only squares. The following are samples of the resulting equivalent theorems: If r =4m- I or In= I, • • • , 7, then rep resents all positive multiples of 4. If k =1, • • • , 6, and s = o or 1, then - represents all positive multiples of 12. If 3 < k 22, s=o or 1, o MiC. 5, with M+s even, then I I (44k - Ls - represents all positive mul tiples of 44.