mathematics

Wieferich pair

Wieferich pair 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 Wieferich pair rather than just read about it. In short: In mathematics, a Wieferich pair is a pair of prime numbers p and q that satisfy pq − 1 ≡ 1 (mod q2) and qp − 1 ≡ 1 (mod p2) Wieferich pairs are named after German mathematician Arthur Wieferich. Wieferich pairs play an important role in Preda Mihăilescu's 2002 proof of Mihăilescu's theorem (formerly known as Catalan's conjecture).

Key takeaways

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

Reference excerpt

In mathematics, a Wieferich pair is a pair of prime numbers p and q that satisfy

pq − 1 ≡ 1 (mod q2) and qp − 1 ≡ 1 (mod p2) Wieferich pairs are named after German mathematician Arthur Wieferich. Wieferich pairs play an important role in Preda Mihăilescu's 2002 proof of Mihăilescu's theorem (formerly known as Catalan's conjecture).

Known Wieferich pairs There are only 7 Wieferich pairs known:

(2, 1093), (3, 1006003), (5, 1645333507), (5, 188748146801), (83, 4871), (911, 318917), and (2903, 18787). (sequence OEIS: A124121 and OEIS: A124122 in OEIS)

Wieferich triple A Wieferich triple is a triple of prime numbers p, q and r that satisfy

pq − 1 ≡ 1 (mod q2), qr − 1 ≡ 1 (mod r2), and rp − 1 ≡ 1 (mod p2). There are 17 known Wieferich triples:

(2, 1093, 5), (2, 3511, 73), (3, 11, 71), (3, 1006003, 3188089), (5, 20771, 18043), (5, 20771, 950507), (5, 53471161, 193), (5, 6692367337, 1601), (5, 6692367337, 1699), (5, 188748146801, 8807), (13, 863, 23), (17, 478225523351, 2311), (41, 138200401, 2953), (83, 13691, 821), (199, 1843757, 2251), (431, 2393, 54787), and (1657, 2281, 1667). (sequences OEIS: A253683, OEIS: A253684 and OEIS: A253685 in OEIS)

Barker sequence Barker sequence or Wieferich n-tuple is a generalization of Wieferich pair and Wieferich triple. It is primes (p1, p2, p3, ..., pn) such that

p1p2 − 1 ≡ 1 (mod p22), p2p3 − 1 ≡ 1 (mod p32), p3p4 − 1 ≡ 1 (mod p42), ..., pn−1pn − 1 ≡ 1 (mod pn2), pnp1 − 1 ≡ 1 (mod p12). For example, (3, 11, 71, 331, 359) is a Barker sequence, or a Wieferich 5-tuple; (5, 188748146801, 453029, 53, 97, 76704103313, 4794006457, 12197, 3049, 41) is a Barker sequence, or a Wieferich 10-tuple. For the smallest Wieferich n-tuple, see (sequence A271100 in the OEIS), for the ordered set of all Wieferich tuples, see (sequence A317721 in the OEIS).

Wieferich sequence Wieferich sequence is a special type of Barker sequence. Every integer k>1 has its own Wieferich sequence. To make a Wieferich sequence of an integer k>1, start with a(1)=k, a(n) = the smallest prime p such that a(n−1)p−1 = 1 (mod p) but a(n−1) ≠ 1 or −1 (mod p). It is a conjecture that every integer k>1 has a periodic Wieferich sequence. For example, the Wieferich sequence of 2:

2, 1093, 5, 20771, 18043, 5, 20771, 18043, 5, ..., it gets a cycle: {5, 20771, 18043}. (a Wieferich triple) The Wieferich sequence of 83:

83, 4871, 83, 4871, 83, 4871, 83, ..., it gets a cycle: {83, 4871}. (a Wieferich pair) The Wieferich sequence of 59: (this sequence needs more terms to be periodic)

59, 2777, 133287067, 13, 863, 7, 5, 20771, 18043, 5, ... it also gets 5. However, there are many values of a(1) with unknown status. For example, the Wieferich sequence of 3:

3, 11, 71, 47, ? (There are no known Wieferich primes in base 47). The Wieferich sequence of 14:

14, 29, ? (There are no known Wieferich primes in base 29 except 2, but 22 = 4 divides 29 − 1 = 28) The Wieferich sequence of 39:

39, 8039, 617, 101, 1050139, 29, ? (It also gets 29) It is unknown that values for k exist such that the Wieferich sequence of k does not become periodic. Eventually, it is unknown that values for k exist such that the Wieferich sequence of k is finite. When a(n − 1)=k, a(n) will be (start with k = 2): 1093, 11, 1093, 20771, 66161, 5, 1093, 11, 487, 71, 2693, 863, 29, 29131, 1093, 46021, 5, 7, 281, ?, 13, 13, 25633, 20771, 71, 11, 19, ?, 7, 7, 5, 233, 46145917691, 1613, 66161, 77867, 17, 8039, 11, 29, 23, 5, 229, 1283, 829, ?, 257, 491531, ?, ... (For k = 21, 29, 47, 50, even the next value is unknown)

See also Wieferich prime Fermat quotient

References

Further reading Bilu, Yuri F. (2004). "Catalan's conjecture (after Mihăilescu)". Astérisque. 294: vii, 1–26. Zbl 1094.11014. Ernvall, Reijo; Metsänkylä, Tauno (1997). "On the p-divisibility of Fermat quotients". Math. Comp. 66 (219): 1353–1365. Bibcode:1997MaCom..66.1353E. doi:10.1090/S0025-5718-97-00843-0. MR 1408373. Zbl 0903.11002. Steiner, Ray (1998). "Class number bounds and Catalan's equation". Math. Comp. 67 (223): 1317–1322. Bibcode:1998MaCom..67.1317S. doi:10.1090/S0025-5718-98-00966-1. MR 1468945. Zbl 0897.11009.

Worked examples

Example 1 — a first encounter with Wieferich pair

Start with the simplest possible case. Write down what Wieferich pair 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 Wieferich pair 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 Wieferich pair 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 Wieferich pair

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

Read the reference excerpt below once, without taking notes.
Close the page and write down what Wieferich pair 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.
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Frequently asked questions

What is Wieferich pair in simple terms?

In mathematics, a Wieferich pair is a pair of prime numbers p and q that satisfy pq − 1 ≡ 1 (mod q2) and qp − 1 ≡ 1 (mod p2) Wieferich pairs are named after German mathematician Arthur Wieferich. Wieferich pairs play an important role in Preda Mihăilescu's 2002 proof of Mihăilescu's theorem (former…

Why does Wieferich pair 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 Wieferich pair?

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 Wieferich pair.