The number of representations of any positive integer N as a sum of four squares is 8s or 24s, according as N is odd or even, where s is the sum of the positive odd divisors of N. If N is odd, the number of representations of 4N as a sum of four odd squares is 16s.
Analogous theorems have been found for many forms whose four coefficients are all products of powers of 2 and 3, or when each coefficient is I or 5. Also the number of proper representa tions was found.
Later writers gradually reduced this limit 53 to 37. The latest result is that every p is a sum of 17 biquadrates and io doubles of biquadrates.
That every p is a sum of nine integral cubes > o was first proved by Wieferich in 1909 a gap in his proof was first filled by Kempner in 1912. A simpler proof was given by Dickson in 1927, who proved also that, if 23, every p is represented by C8, where C8 is a sum of 8 integral cubes o, and x > o also by C7 if k 34, kio, 15, 20, 25, 3o; also by if / < 9, /0 5 and for p<4o,000 by In 1909 Hilbert proved that every p is a sum of Nk positive or zero kth powers. The long proof gives no clue as to the value of the finite number Nk. By means of a fivefold integral, he proved the existence of an identity, of type (I), which expresses k as a sum of 2kth powers of linear func tions of x2, x3, with integral coefficients. Here Al is a positive integer depending on k. Later writers established this identity by algebraic methods.
a sum of r sixth powers and the products of s sixth powers by 8, Dickson proved that every p is represented by each of the forms Identities of Hilbert's type leading to explicit finite values of Nk were given by A. Hurwitz for k= 8, by I. Schur for k= io, and by Kempner for k= 12 and 14.
By intricate analytic investigations, Hardy and Littlewood recently obtained remarkable results. First they gave a new proof that Nk is finite. Next, they proved that, for all sufficiently large numbers, Nk-(k- (9 cubes, 21 biquadrates, 53 fifth powers, etc.). Finally, when k 4 they obtained in 1925 a limit involving logarithms which gives the best results to date: Every sufficiently large integer p is a sum of 19 bi quadrates, 41 fifth powers, 87 sixth powers, 193 seventh powers, 425 eighth powers, 949 ninth powers, or 2,113 tenth powers. No determination was made for the limit beyond which p must lie, but it would be excessively large. Landau simplified this theory in his 1927 text.
In 1909 Landau proved that all sufficiently large integers p are sums of 8 cubes. That this holds if 23X Io" was proved by Baer by technical results in the analytic theory of primes.
For 1=1, , 5, Dickson proved that represents all sufficiently large integers.