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 in 1938.
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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, ..., pt}, such that N = p1k + p2k + ... + ptk, where t is at most some constant value?
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 x1/k/log x. From this, the number of t-term expressions with sums ≤x is roughly xt/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, using and refining the methods of Hardy, Litllewood and Vinogradov, Hua Luogeng obtains a O(k2log 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.