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.

Contents

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

In his monograph[3], using and refining the methods of Hardy, Litllewood and Vinogradov, Hua Logeng 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.[4]

References

  1. ^ L. K. Hua: Some results in additive prime number theory, Quart. J. Math. Oxford, 9(1938), 68–80.
  2. ^ Buttcane, Jack (January 2010). "A note on the Waring–Goldbach problem". Journal of Number Theory (Elsevier) 130 (1): 116–127. doi:10.1016/j.jnt.2009.07.006. 
  3. ^ Hua Lo Keng: Additive theory of prime numbers, Translations of Mathematical Monographs, 13, American Mathematical Society, Providence, R.I. 1965 xiii+190 pp
  4. ^ Kawada, Koichi; Wooley, Trevor D. (2001), "On the Waring–Goldbach problem for fourth and fifth powers", Proceedings of the London Mathematical Society 83 (1): 1–50, doi:10.1112/plms/83.1.1 .