Lucas's theorem

Key takeaways

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

Reference excerpt

In number theory, Lucas's theorem expresses the remainder of division of the binomial coefficient ( m n ) {\displaystyle {\tbinom {m}{n}}} by a prime number p in terms of the base p expansions of the integers m and n. Lucas's theorem first appeared in 1878 in papers by Édouard Lucas.

Statement For non-negative integers m and n and a prime p, the following congruence relation holds:

( m n ) ≡ ∏ i = 0 k ( m i n i ) ( mod p ) , {\displaystyle {\binom {m}{n}}\equiv \prod _{i=0}^{k}{\binom {m_{i}}{n_{i}}}{\pmod {p}},}

where

m = m k p k + m k − 1 p k − 1 + ⋯ + m 1 p + m 0 , {\displaystyle m=m_{k}p^{k}+m_{k-1}p^{k-1}+\cdots +m_{1}p+m_{0},}

and

n = n k p k + n k − 1 p k − 1 + ⋯ + n 1 p + n 0 {\displaystyle n=n_{k}p^{k}+n_{k-1}p^{k-1}+\cdots +n_{1}p+n_{0}}

are the base p expansions of m and n respectively. This uses the convention that ( m n ) = 0 {\displaystyle {\tbinom {m}{n}}=0} if m < n.

Proofs There are several ways to prove Lucas's theorem.

Consequences One consequence of Lucas's theorem is that the binomial coefficient ( m n ) {\displaystyle {\tbinom {m}{n}}} is divisible by the prime p if and only if at least one of the digits of the base-p representation of n is greater than the corresponding digit of m. In particular, ( m n ) {\displaystyle {\tbinom {m}{n}}} is odd if and only if the positions of the ones in the binary expansion of n are a subset of the positions of the ones in that of m. This leads to a peculiar distribution of odd numbers in Pascal's triangle, resembling Sierpiński 's triangle, shown to the right.

Non-prime moduli Lucas's theorem can be generalized to give an expression for the remainder when ( m n ) {\displaystyle {\tbinom {m}{n}}} is divided by a prime power pk. However, the formulas become more complicated. If the modulus is the square of a prime p, the following congruence relation holds for all 0 ≤ s ≤ r ≤ p − 1, a ≥ 0, and b ≥ 0:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Lucas's theorem

Start with the simplest possible case. Write down what Lucas's theorem 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's theorem 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's theorem 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's theorem

In research: Lucas's theorem 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's theorem 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's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theorems about prime numbers, so understanding it makes those chapters shorter.
In everyday life: Look for Lucas's theorem 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.

How to study Lucas's theorem in 20 minutes

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

Frequently asked questions

What is Lucas's theorem in simple terms?

In number theory, Lucas's theorem expresses the remainder of division of the binomial coefficient ( m n ) {\displaystyle {\tbinom {m}{n}}} by a prime number p in terms of the base p expansions of the integers m and n. Lucas's theorem first appeared in 1878 in papers by Édouard Lucas.

Why does Lucas's theorem 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's theorem?

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's theorem.