mathematics

Lemoine's conjecture

Lemoine's conjecture 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 Lemoine's conjecture rather than just read about it. In short: In number theory, Lemoine's conjecture, also sometimes known as Levy's conjecture, states that all odd integers greater than 5 can be represented as the sum of an odd prime number and an even semiprime. The conjecture was first proposed by Émile Lemoine in 1895, but was erroneously attributed by MathWorld to Hyman Levy who pondered it in the 1960s.

Key takeaways

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

Reference excerpt

In number theory, Lemoine's conjecture, also sometimes known as Levy's conjecture, states that all odd integers greater than 5 can be represented as the sum of an odd prime number and an even semiprime. The conjecture was first proposed by Émile Lemoine in 1895, but was erroneously attributed by MathWorld to Hyman Levy who pondered it in the 1960s.

Formal definition Algebraically, the conjecture states that 2 n + 1 = p + 2 q {\displaystyle 2n+1=p+2q} always has a solution in primes p {\displaystyle p} and q {\displaystyle q} (not necessarily distinct) for n > 2 {\displaystyle n>2} . The Lemoine conjecture implies Goldbach's weak conjecture. In 2008, Zhi-Wei Sun similarly conjectured that all odd integers greater than 3 can be represented as the sum of a prime number and the product of two consecutive positive integers; that is, 2 n + 1 = p + m ( m + 1 ) {\displaystyle 2n+1=p+m(m+1)} for all n > 1 {\displaystyle n>1} .

Example For example, the odd integer 47 {\displaystyle 47} can be expressed as the sum of a prime and an even semiprime in four different ways:

47 = 13 + 2 ⋅ 17 = 37 + 2 ⋅ 5 = 41 + 2 ⋅ 3 = 43 + 2 ⋅ 2 {\displaystyle 47=13+2\cdot 17=37+2\cdot 5=41+2\cdot 3=43+2\cdot 2}

The number of ways this can be done is given by (sequence A046927 in the OEIS). Lemoine's conjecture is that this sequence contains no zeros after the first three.

Evidence According to MathWorld, the conjecture has been verified by Corbitt up to 109. A blog post in June of 2019 additionally claimed to have verified the conjecture up to 1010. A proof was claimed in 2017 by Agama and Gensel, but this was later found to be flawed.

See also Lemoine's conjecture and extensions

Notes

References Emile Lemoine, L'intermédiaire des mathématiciens, 1 (1894), 179; ibid 3 (1896), 151. H. Levy, "On Goldbach's Conjecture", Math. Gaz. 47 (1963): 274 L. Hodges, "A lesser-known Goldbach conjecture", Math. Mag., 66 (1993): 45–47. doi:10.2307/2690477. JSTOR 2690477 John O. Kiltinen and Peter B. Young, "Goldbach, Lemoine, and a Know/Don't Know Problem", Mathematics Magazine, 58(4) (Sep., 1985), pp. 195–203. doi:10.2307/2689513. JSTOR 2689513 Richard K. Guy, Unsolved Problems in Number Theory New York: Springer-Verlag 2004: C1

External links Levy's Conjecture by Jay Warendorff, Wolfram Demonstrations Project.

Worked examples

Example 1 — a first encounter with Lemoine's conjecture

Start with the simplest possible case. Write down what Lemoine's conjecture 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 Lemoine's conjecture 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 Lemoine's conjecture 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 Lemoine's conjecture

In research: Lemoine's conjecture 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 Lemoine's conjecture 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: Lemoine's conjecture 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 Lemoine's conjecture 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 Lemoine's conjecture in 20 minutes

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

Frequently asked questions

What is Lemoine's conjecture in simple terms?

Why does Lemoine's conjecture 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 Lemoine's conjecture?

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 Lemoine's conjecture.