mathematics

Waring–Goldbach problem

Waring–Goldbach problem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Waring–Goldbach problem rather than just read about it. In short: 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.

Key takeaways

Waring–Goldbach problem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Waring–Goldbach problem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Waring–Goldbach problem from memory before moving on to harder problems.

Reference excerpt

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 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, ..., pt}, such that N = p1k + p2k + ... + ptk, 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 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, 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.

References

Worked examples

Example 1 — a first encounter with Waring–Goldbach problem

Start with the simplest possible case. Write down what Waring–Goldbach problem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Waring–Goldbach problem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Waring–Goldbach problem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Waring–Goldbach problem

In research: Waring–Goldbach problem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Waring–Goldbach problem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Waring–Goldbach problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Additive number theory, Conjectures about prime numbers, Unsolved problems in number theory, so understanding it makes those chapters shorter.
In everyday life: Look for Waring–Goldbach problem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Waring–Goldbach problem” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Waring–Goldbach problem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Waring–Goldbach problem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Waring–Goldbach problem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Waring–Goldbach problem in simple terms?

Why does Waring–Goldbach problem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Waring–Goldbach problem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Waring–Goldbach problem.