mathematics

Lucas–Carmichael number

Lucas–Carmichael number 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 Lucas–Carmichael number rather than just read about it. In short: In mathematics, a Lucas–Carmichael number is a positive composite integer n such that If p is a prime factor of n, then p + 1 is a factor of n + 1; n is odd and square-free. The first condition resembles Korselt's criterion for Carmichael numbers, where −1 is replaced with +1.

Key takeaways

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

Reference excerpt

In mathematics, a Lucas–Carmichael number is a positive composite integer n such that

If p is a prime factor of n, then p + 1 is a factor of n + 1; n is odd and square-free. The first condition resembles Korselt's criterion for Carmichael numbers, where −1 is replaced with +1. The second condition eliminates from consideration some trivial cases like cubes of prime numbers, such as 8 or 27, which otherwise would be Lucas–Carmichael numbers (since n3 + 1 = (n + 1)(n2 − n + 1) is always divisible by n + 1). They are named after Édouard Lucas and Robert Carmichael. The first few Lucas–Carmichael numbers are:

399, 935, 2015, 2915, 4991, 5719, 7055, 8855, 12719, 18095, 20705... (sequence A006972 in the OEIS)

Properties The smallest Lucas–Carmichael number is 399 = 3 × 7 × 19. It is easy to verify that 3+1, 7+1, and 19+1 are all factors of 399+1 = 400. The smallest Lucas–Carmichael number with 4 factors is 8855 = 5 × 7 × 11 × 23. The smallest Lucas–Carmichael number with 5 factors is 588455 = 5 × 7 × 17 × 23 × 43. It is not known whether any Lucas–Carmichael number is also a Carmichael number. Thomas Wright proved in 2016 that there are infinitely many Lucas–Carmichael numbers. If we let N ( X ) {\displaystyle N(X)} denote the number of Lucas–Carmichael numbers up to X {\displaystyle X} , Wright showed that there exists a positive constant K {\displaystyle K} such that

N ( X ) ≫ X K / ( log ⁡ log ⁡ log ⁡ X ) 2 {\displaystyle N(X)\gg X^{K/\left(\log \log \log X\right)^{2}}} .

References

External links Richard Guy (2004). "Section A13". Unsolved Problems in Number Theory (3rd ed.). Springer Verlag. Lucas–Carmichael number at PlanetMath. "Something special about 399 (and 2015) - Numberphile". YouTube. 15 January 2015. Archived from the original on 2021-12-22.

Worked examples

Example 1 — a first encounter with Lucas–Carmichael number

Start with the simplest possible case. Write down what Lucas–Carmichael number 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 Lucas–Carmichael number 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 Lucas–Carmichael number 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 Lucas–Carmichael number

In research: Lucas–Carmichael number 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 Lucas–Carmichael number 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: Lucas–Carmichael number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Integer sequences, so understanding it makes those chapters shorter.
In everyday life: Look for Lucas–Carmichael number 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 Lucas–Carmichael number in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Lucas–Carmichael number 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 Lucas–Carmichael number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lucas–Carmichael number in simple terms?

In mathematics, a Lucas–Carmichael number is a positive composite integer n such that If p is a prime factor of n, then p + 1 is a factor of n + 1; n is odd and square-free. The first condition resembles Korselt's criterion for Carmichael numbers, where −1 is replaced with +1.

Why does Lucas–Carmichael number 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 Lucas–Carmichael number?

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 Lucas–Carmichael number.