Waring–Goldbach problem
The Waring–Goldbach problem is a problem in
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?[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 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
This section needs expansion with: Published results that are very similar, or that are likely to contribute to its eventual proof. You can help by adding to it. (March 2010) |
In his monograph,
Every sufficiently large odd integer is the sum of 21 fifth powers of primes.[4]
References
- ^ L. K. Hua: Some results in additive prime number theory, Quart. J. Math. Oxford, 9(1938), 68–80.
- .
- ^ Hua Lo Keng: Additive theory of prime numbers, Translations of Mathematical Monographs, 13, American Mathematical Society, Providence, R.I. 1965 xiii+190 pp
- hdl:2027.42/135164.