Definition
A Wieferich pair is an ordered pair of distinct prime numbers $(p,q)$ that satisfy the simultaneous congruences
$$ p^{,q-1}\equiv 1 \pmod{q^{2}} \qquad\text{and}\qquad q^{,p-1}\equiv 1 \pmod{p^{2}} . $$
In other words, each prime behaves as a Wieferich prime with respect to the other prime taken as the base of the exponentiation.
Overview
The concept generalises the notion of a Wieferich prime, which is a single prime $r$ for which
$$ a^{,r-1}\equiv 1 \pmod{r^{2}} $$
holds for a fixed integer base $a$ (most commonly $a=2$).
A Wieferich pair therefore represents a mutual relationship: the first prime is a Wieferich prime to the base given by the second prime, and vice‑versa. The study of such pairs lies within elementary and computational number theory and is connected to open problems such as the distribution of Wieferich primes and the abc‑conjecture.
Etymology / Origin
The term is named after the 19th‑century German mathematician Arthur Wieferich, who first investigated primes $p$ satisfying $2^{p-1}\equiv1\pmod{p^{2}}$ in connection with Fermat’s Last Theorem. The extension to pairs of primes emerged in later research on reciprocal congruence conditions; the phrase “Wieferich pair” appears in the mathematical literature from the early 2000s.
Characteristics
| Property | Description |
|---|---|
| Symmetry | The definition is symmetric: $(p,q)$ is a Wieferich pair iff $(q,p)$ is. |
| Trivial families | If $a$ is a prime and $p$ is a Wieferich prime to base $a$ (i.e., $a^{p-1}\equiv1\pmod{p^{2}}$), then $(a,p)$ forms a Wieferich pair because $p^{a-1}\equiv p^{a-1}\equiv 1\pmod{a^{2}}$ is automatically satisfied when $a$ is small (e.g., $a=2$ or $a=3$). |
| Known examples | The only experimentally verified pairs (as of the latest computational searches) are: • $(2,1093)$ and $(2,3511)$ – both involve the two known base‑2 Wieferich primes. • $(3,1006003)$ – involving the known base‑3 Wieferich prime. No other pairs have been discovered up to the limits of exhaustive searches (currently beyond $10^{12}$ for the smaller component). |
| Search algorithms | Verification of a candidate pair requires modular exponentiation with moduli $p^{2}$ and $q^{2}$. Efficient algorithms exploit repeated squaring and the Chinese Remainder Theorem; large‑scale searches are distributed across computing clusters. |
| Open questions | It is unknown whether infinitely many Wieferich pairs exist, whether there exist pairs where neither component is a small base (e.g., neither 2 nor 3), or whether any pair can involve two non‑trivial Wieferich primes (primes that are Wieferich for a base different from the other component). |
Related Topics
- Wieferich prime – a single prime satisfying a single-base congruence.
- Fermat’s Last Theorem – historical motivation for studying Wieferich primes.
- Fermat quotients – the integer $\frac{a^{p-1}-1}{p}$ whose divisibility by $p$ characterises Wieferich primes.
- abc conjecture – conjectural implications for the rarity of Wieferich primes and, by extension, Wieferich pairs.
- Modular exponentiation – computational technique central to testing the defining congruences.
- Pseudoprimes and Carmichael numbers – broader classes of composite numbers satisfying Fermat‑type congruences, providing context for the special nature of Wieferich phenomena.