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 LuogengL. K. Hua: Some results in additive prime number theory, Quart. J. Math. Oxford, 9(1938), 68–80. 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 positive integer , is it true that all sufficiently large numbers can be written as , where are primes and is at most some constant value?
The case is a weak version of the Goldbach conjecture. Some progress has been made for .
Heuristic justification By the prime number theorem, the number of th powers of a prime below is of the order . From this, heuristically, the number of sums of terms which are at most is roughly . It is reasonable to assume that for some sufficiently large number this is , i.e., all numbers up to are -fold sums of th powers of primes. This argument is, of course, a long way from a strict proof.
Relevant results
In his monograph,Hua Lo Keng: Additive theory of prime numbers, Translations of Mathematical Monographs, 13, American Mathematical Society, Providence, R.I. 1965 xiii+190 pp using and refining the methods of Hardy, Littlewood and Vinogradov, Hua Luogeng obtained upper bound for the number of terms required to exhibit all sufficiently large numbers as the sum of th powers of primes.
Every sufficiently large odd integer is the sum of 21 fifth powers of primes..
References