mathematics

Lehmer's totient problem

Lehmer's totient 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 Lehmer's totient problem rather than just read about it. In short: In mathematics, Lehmer's totient problem asks whether there is any composite number n such that Euler's totient function φ(n) divides n − 1. This is an unsolved problem.

Key takeaways

Lehmer's totient 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 Lehmer's totient problem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Lehmer's totient problem from memory before moving on to harder problems.

Reference excerpt

In mathematics, Lehmer's totient problem asks whether there is any composite number n such that Euler's totient function φ(n) divides n − 1. This is an unsolved problem. It is known that φ(n) = n − 1 if and only if n is prime. So for every prime number n, we have φ(n) = n − 1 and thus in particular φ(n) divides n − 1. D. H. Lehmer asked in 1932 whether there exist composite numbers with this property.

History Lehmer showed that if any composite solution n exists, it must be odd, square-free, and divisible by at least seven distinct primes (i.e. ω(n) ≥ 7). Such a number must also be a Carmichael number. In 1980, Cohen and Hagis proved that, for any solution n to the problem, n > 1020 and ω(n) ≥ 14. In 1988, Hagis showed that if 3 divides any solution n, then n > 101‍937‍042 and ω(n) ≥ 298848. This was subsequently improved by Burcsi, Czirbusz, and Farkas, who showed that if 3 divides any solution n, then n > 10360‍000‍000 and ω(n) ≥ 40‍000‍000. A 2011 result of Luca and Pomerance states that the number of solutions to the problem less than X is at most X1/2 / (log X)1/2 + o(1). In 2019 Burek and Żmija proved that any solution n satisfies n < 22ω(n) - 22ω(n)-1.

References

Worked examples

Example 1 — a first encounter with Lehmer's totient problem

Start with the simplest possible case. Write down what Lehmer's totient 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 Lehmer's totient 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 Lehmer's totient 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 Lehmer's totient problem

In research: Lehmer's totient 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 Lehmer's totient 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: Lehmer's totient problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Conjectures, Multiplicative functions, Unsolved problems in number theory, so understanding it makes those chapters shorter.
In everyday life: Look for Lehmer's totient 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.

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How to study Lehmer's totient problem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Lehmer's totient 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 Lehmer's totient problem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lehmer's totient problem in simple terms?

In mathematics, Lehmer's totient problem asks whether there is any composite number n such that Euler's totient function φ(n) divides n − 1. This is an unsolved problem.

Why does Lehmer's totient 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 Lehmer's totient 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 Lehmer's totient problem.