The Waring–Goldbach problem is a problem in
additive number theory
Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigro ...
, concerning the representation of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s as sums of powers of
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
s. It is named as a combination of
Waring's problem
In number theory, Waring's problem asks whether each natural number ''k'' has an associated positive integer ''s'' such that every natural number is the sum of at most ''s'' natural numbers raised to the power ''k''. For example, every natural numb ...
on sums of powers of integers, and the
Goldbach conjecture
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.
The conjecture has been shown to hold ...
on sums of primes. It was initiated by
Hua Luogeng 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, , such that ''N'' = ''p''
1''k'' + ''p''
2''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, 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, Littlewood and
Vinogradov, Hua Luogeng obtains a ''O''(''k''
2log ''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.
[.]
References
{{DEFAULTSORT:Waring-Goldbach problem
Additive number theory
Conjectures about prime numbers
Unsolved problems in number theory