The Waring–Goldbach problem is a problem in additive number theory

Additive number theory

In number theory, the specialty additive number theory studies subsets of integers and their behavior under addition. More abstractly, the field of "additive number theory" includes the study of Abelian groups and commutative semigroups with an operation of addition. Additive number theory has...

, concerning the representation of integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s as sums of powers of prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s. It is named as a combination of Waring's problem

Waring's problem

In number theory, Waring's problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers...

 on sums of powers of integers, and the Goldbach conjecture on sums of primes. It was initiated by Hua Luogeng

Hua Luogeng

Hua Luogeng was a Chinese mathematician born in Jintan, Jiangsu. He was the founder and pioneer in many fields in mathematical research. He wrote more than 200 papers and monographs, many of which became classics. Since his sudden death while delivering a lecture at the University of Tokyo, Japan,...

 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 natural number, k, is it true that for sufficiently large integer N there necessarily exist a set of primes, {p₁, p₂, ..., p_t}, such that N = p₁^k + p₂^k + ... + p_t^k, 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

Prime number theorem

In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....

, the number of k-th powers of a prime below x is of the order x^1/k/log x.
From this, the number of t-term expressions with sums ≤x is roughly x^t/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

Ivan Matveyevich Vinogradov

Ivan Matveevich Vinogradov was a Soviet mathematician, who was one of the creators of modern analytic number theory, and also a dominant figure in mathematics in the USSR. He was born in the Velikiye Luki district, Pskov Oblast. He graduated from the University of St...

, 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.