mathematics

Wolstenholme's theorem

Wolstenholme's theorem 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 Wolstenholme's theorem rather than just read about it. In short: In mathematics, Wolstenholme's theorem states that for a prime number p ≥ 5, the congruence ( 2 p − 1 p − 1 ) ≡ 1 ( mod p 3 ) {\displaystyle {2p-1 \choose p-1}\equiv 1{\pmod {p^{3}}}} holds, where the parentheses denote a binomial coefficient. For example, with p = 7, this says that 1716 is one more than a multiple of 343.

Key takeaways

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

Reference excerpt

In mathematics, Wolstenholme's theorem states that for a prime number p ≥ 5, the congruence

( 2 p − 1 p − 1 ) ≡ 1 ( mod p 3 ) {\displaystyle {2p-1 \choose p-1}\equiv 1{\pmod {p^{3}}}}

holds, where the parentheses denote a binomial coefficient. For example, with p = 7, this says that 1716 is one more than a multiple of 343. The theorem was first proved by Joseph Wolstenholme in 1862. In 1819, Charles Babbage showed the same congruence modulo p2, which holds for p ≥ 3. An equivalent formulation is the congruence

( a p b p ) ≡ ( a b ) ( mod p 3 ) {\displaystyle {ap \choose bp}\equiv {a \choose b}{\pmod {p^{3}}}}

for p ≥ 5, which is due to Wilhelm Ljunggren (and, in the special case b = 1, to J. W. L. Glaisher) and is inspired by Lucas's theorem. No known composite numbers satisfy Wolstenholme's theorem and it is conjectured that there are none (see below). A prime that satisfies the congruence modulo p4 is called a Wolstenholme prime (see below). As Wolstenholme himself established, his theorem can also be expressed as a pair of congruences for (generalized) harmonic numbers:

1 + 1 2 + 1 3 + ⋯ + 1 p − 1 ≡ 0 ( mod p 2 ) , and {\displaystyle 1+{1 \over 2}+{1 \over 3}+\dots +{1 \over p-1}\equiv 0{\pmod {p^{2}}}{\mbox{, and}}}

1 + 1 2 2 + 1 3 2 + ⋯ + 1 ( p − 1 ) 2 ≡ 0 ( mod p ) . {\displaystyle 1+{1 \over 2^{2}}+{1 \over 3^{2}}+\dots +{1 \over (p-1)^{2}}\equiv 0{\pmod {p}}.}

since

( 2 p − 1 p − 1 ) = ∏ 1 ≤ k ≤ p − 1 2 p − k k ≡ 1 − 2 p ∑ 1 ≤ k ≤ p − 1 1 k ( mod p 2 ) {\displaystyle {2p-1 \choose p-1}=\prod _{1\leq k\leq p-1}{\frac {2p-k}{k}}\equiv 1-2p\sum _{1\leq k\leq p-1}{\frac {1}{k}}{\pmod {p^{2}}}}

(Congruences with fractions make sense, provided that the denominators are coprime to the modulus.) For example, with p = 7, the first of these says that the numerator of 49/20 is a multiple of 49, while the second says the numerator of 5369/3600 is a multiple of 7.

Wolstenholme primes

A prime p is called a Wolstenholme prime iff the following condition holds:

( 2 p − 1 p − 1 ) ≡ 1 ( mod p 4 ) . {\displaystyle {{2p-1} \choose {p-1}}\equiv 1{\pmod {p^{4}}}.}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Wolstenholme's theorem

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

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

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Wolstenholme's theorem” →

Affiliate

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

How to study Wolstenholme's theorem in 20 minutes

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

Frequently asked questions

What is Wolstenholme's theorem in simple terms?

In mathematics, Wolstenholme's theorem states that for a prime number p ≥ 5, the congruence ( 2 p − 1 p − 1 ) ≡ 1 ( mod p 3 ) {\displaystyle {2p-1 \choose p-1}\equiv 1{\pmod {p^{3}}}} holds, where the parentheses denote a binomial coefficient. For example, with p = 7, this says that 1716 is one mor…

Why does Wolstenholme'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 Wolstenholme'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 Wolstenholme's theorem.