Waring–Goldbach problem

The Waring–Goldbach problem is a problem in additive number theory, concerning the representation of integers as sums of powers of prime numbers. It is named as a combination of Waring's problem on sums of powers of integers, and the Goldbach conjecture on sums of primes. It was initiated by Hua Luogeng^[1] in 1938.

Problem statement

It asks whether large numbers can be expressed as a sum, with at most a constant number of terms, of like powers of primes. That is, for any given natural number, k, is it true that for sufficiently large integer N there necessarily exist a set of primes, {p₁, p₂, ..., p_t}, such that N = p₁^k + p₂^k + ... + p_t^k, where t is at most some constant value?^[2]

The case, k = 1, is a weaker version of the Goldbach conjecture. Some progress has been made on the cases k = 2 to 7.

Heuristic justification

By the prime number theorem, the number of k-th powers of a prime below x is of the order x^1/k/log x. From this, the number of t-term expressions with sums ≤x is roughly x^t/k/(log x)^t. It is reasonable to assume that for some sufficiently large number t this is x − c, i.e., all numbers up to x are t-fold sums of k-th powers of primes. This argument is, of course, a long way from a strict proof.

Relevant results

In his monograph,^[3] using and refining the methods of Hardy, Littlewood and Vinogradov, Hua Luogeng obtains a O(k² log k) upper bound for the number of terms required to exhibit all sufficiently large numbers as the sum of k-th powers of primes.

Every sufficiently large odd integer is the sum of 21 fifth powers of primes.^[4]