mathematics

Sierpiński number

Sierpiński 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 Sierpiński number rather than just read about it. In short: In number theory, a Sierpiński number is an odd natural number k such that k × 2 n + 1 {\displaystyle k\times 2^{n}+1} is composite for all natural numbers n. In 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which have this property.

Key takeaways

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

Reference excerpt

In number theory, a Sierpiński number is an odd natural number k such that k × 2 n + 1 {\displaystyle k\times 2^{n}+1} is composite for all natural numbers n. In 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which have this property. In other words, when k is a Sierpiński number, all members of the following set are composite:

{ k ⋅ 2 n + 1 : n ∈ N } . {\displaystyle \left\{\,k\cdot 2^{n}+1:n\in \mathbb {N} \,\right\}.}

If the form is instead k × 2 n − 1 {\displaystyle k\times 2^{n}-1} , then k is a Riesel number.

Known Sierpiński numbers The sequence of currently known Sierpiński numbers begins with:

78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, 965431, 1259779, 1290677, 1518781, 1624097, 1639459, 1777613, 2131043, 2131099, 2191531, 2510177, 2541601, 2576089, 2931767, 2931991, ... (sequence A076336 in the OEIS). The number 78557 was proved to be a Sierpiński number by John Selfridge in 1962, who showed that all numbers of the form 78557⋅2n + 1 have a factor in the covering set {3, 5, 7, 13, 19, 37, 73}. For another known Sierpiński number, 271129, the covering set is {3, 5, 7, 13, 17, 241}. Most currently known Sierpiński numbers possess similar covering sets. However, in 1995 A. S. Izotov showed that some fourth powers could be proved to be Sierpiński numbers without establishing a covering set for all values of n. His proof depends on the aurifeuillean factorization t4⋅24m+2 + 1 = (t2⋅22m+1 + t⋅2m+1 + 1)⋅(t2⋅22m+1 − t⋅2m+1 + 1). This establishes that all n ≡ 2 (mod 4) give rise to a composite, and so it remains to eliminate only n ≡ 0, 1, 3 (mod 4) using a covering set.

Smallest Sierpiński number

The Sierpiński problem asks for the value of the smallest Sierpiński number. In private correspondence with Paul Erdős, Selfridge conjectured that 78,557 was the smallest Sierpiński number. No smaller Sierpiński numbers have been discovered, and it is now believed that 78,557 is the smallest number. To show that 78,557 really is the smallest Sierpiński number, one must show that all the odd numbers smaller than 78,557 are not Sierpiński numbers. That is, for every odd k below 78,557, there needs to exist a positive integer n such that k2n + 1 is prime. The distributed volunteer computing project PrimeGrid is attempting to eliminate all the remaining values of k:

k = 21181, 22699, 24737, 55459, and 67607. The current status for the remaining multipliers can be seen at PrimeGrid's website.

Smallest prime Sierpiński number

In 1976, Nathan Mendelsohn determined that the prime number 271,129 is a Sierpiński number. It is the second-smallest known Sierpiński number, and the smallest known prime Sierpiński number, but it is unknown whether others might be smaller. The prime Sierpiński problem asks for the value of the smallest prime Sierpiński number, and there is an ongoing "Prime Sierpiński search" which tries to prove that 271129 is the first Sierpiński number which is also a prime.

Extended Sierpiński problem

Suppose that 78,557 is the smallest Sierpiński number and 271,129 is the smallest prime Sierpiński number. This leaves the second Sierpinski number unknown: there could exist a composite Sierpiński number between 78,557 and 271,129. An ongoing search is trying to prove that 271,129 is the second Sierpiński number by testing all integers in that range, prime or not.

Simultaneously Sierpiński and Riesel A number that is both Sierpiński and Riesel is a Brier number (after Éric Brier). The smallest five known examples are 3316923598096294713661, 10439679896374780276373, 11615103277955704975673, 12607110588854501953787, and 17855036657007596110949 (A076335); it is not known whether other Brier numbers smaller than these exist (i.e., they may not be the five smallest).

Worked examples

Example 1 — a first encounter with Sierpiński number

Start with the simplest possible case. Write down what Sierpiński 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 Sierpiński 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 Sierpiński 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 Sierpiński number

In research: Sierpiński 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 Sierpiński 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: Sierpiński number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Conjectures, Prime numbers, Science and technology in Poland, so understanding it makes those chapters shorter.
In everyday life: Look for Sierpiński 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 Sierpiński number in 20 minutes

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

Frequently asked questions

What is Sierpiński number in simple terms?

Why does Sierpiński 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 Sierpiński 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 Sierpiński number.