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 justificationedit
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 resultsedit
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,[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]
Referencesedit
^L. K. Hua: Some results in additive prime number theory, Quart. J. Math. Oxford, 9(1938), 68–80.
^Buttcane, Jack (January 2010). "A note on the Waring–Goldbach problem". Journal of Number Theory. 130 (1). Elsevier: 116–127. doi:10.1016/j.jnt.2009.07.006.
^Hua Lo Keng: Additive theory of prime numbers, Translations of Mathematical Monographs, 13, American Mathematical Society, Providence, R.I. 1965 xiii+190 pp
^Kawada, Koichi; Wooley, Trevor D. (2001), "On the Waring–Goldbach problem for fourth and fifth powers" (PDF), Proceedings of the London Mathematical Society, 83 (1): 1–50, doi:10.1112/plms/83.1.1, hdl:2027.42/135164.
January 01, 1970
waring, goldbach, problem, problem, additive, number, theory, concerning, representation, integers, sums, powers, prime, numbers, named, combination, waring, problem, sums, powers, integers, goldbach, conjecture, sums, primes, initiated, luogeng, 1938, content. 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 1 Problem statement 2 Heuristic justification 3 Relevant results 4 ReferencesProblem statement editIt 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 editBy 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 editThis 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 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 References edit L K Hua Some results in additive prime number theory Quart J Math Oxford 9 1938 68 80 Buttcane Jack January 2010 A note on the Waring Goldbach problem Journal of Number Theory 130 1 Elsevier 116 127 doi 10 1016 j jnt 2009 07 006 Hua Lo Keng Additive theory of prime numbers Translations of Mathematical Monographs 13 American Mathematical Society Providence R I 1965 xiii 190 pp Kawada Koichi Wooley Trevor D 2001 On the Waring Goldbach problem for fourth and fifth powers PDF Proceedings of the London Mathematical Society 83 1 1 50 doi 10 1112 plms 83 1 1 hdl 2027 42 135164 Retrieved from https en wikipedia org w index php title Waring Goldbach problem amp oldid 1186666730, wikipedia, wiki, book, books, library,
