Wieferich Primes

This video on a special sort of prime piqued my curiosity. It's very well explained. Here is the video:

I asked Gemini to find more about these primes:

To understand Wieferich primes, we first need to look at Fermat’s Little Theorem, which states that for any prime $p$ and any integer $a$ not divisible by $p$:

$$a^{p-1} \equiv 1 \pmod{p}$$

A Wieferich prime takes this a step further. It is a prime number $p$ that satisfies this congruence not just modulo $p$, but modulo $p^2$.

The defining congruence for a generalized Wieferich prime (to base $a$) is:

$$a^{p-1} \equiv 1 \pmod{p^2}$$

Multiplying both sides by $a$, this is sometimes written as $a^p \equiv a \pmod{p^2}$. When mathematicians refer to a "Wieferich prime" without specifying a base, they are universally referring to base 2.

History and Fermat's Last Theorem

The concept was introduced in 1909 by German mathematician Arthur Wieferich. At the time, mathematicians were still fiercely trying to prove Fermat's Last Theorem (FLT), which states that $x^n + y^n = z^n$ has no non-zero integer solutions for $n > 2$.

Wieferich proved a monumental theorem regarding the "First Case" of FLT (where the exponent $p$ does not divide $x$, $y$, or $z$). He proved that if the First Case of Fermat's Last Theorem fails for a prime exponent $p$ (meaning a solution actually exists), then that prime $p$ must satisfy the condition $2^{p-1} \equiv 1 \pmod{p^2}$.

Because Wieferich primes are incredibly rare, this theorem effectively proved the First Case of FLT for almost all prime numbers, long before Andrew Wiles proved the entire theorem in 1994.

Current Status

Wieferich primes are astonishingly rare. Despite massive computational efforts, we only know of exactly two standard (base 2) Wieferich primes:

• 1093 (discovered by Waldemar Meissner in 1913)
• 3511 (discovered by Nicolaas Beeger in 1922)

Distributed computing projects, notably PrimeGrid, have systematically searched for new Wieferich primes. As of recent searches, it has been computationally verified that no other Wieferich primes exist below $6.7 \times 10^{15}$.

Mathematically, their theoretical status remains an open question. It is strongly conjectured that there are infinitely many Wieferich primes, but it has not even been rigorously proven that there is a finite or infinite number of them.

Generalized Wieferich Primes ($a \ge 2$)

We can generalize the definition to check for primes $p$ that satisfy $a^{p-1} \equiv 1 \pmod{p^2}$ for any integer base $a \ge 2$.

When tracking these through integer sequence catalogs like the OEIS, specific bases have their own dedicated entries. Here is a comprehensive list of the known generalized Wieferich primes for the first several bases:

Base ($a$)	Known Wieferich Primes ($p$)	Notes / OEIS Sequence
2	1093, 3511	The "standard" Wieferich primes. (A001220)
3	11, 1006003	Known as Mirimanoff primes. (A014127)
4	1093, 3511	Same as base 2, because $4 = 2^2$.
5	2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801	(A014128)
6	66161, 534851, 3152573, 741584281375, 491531
8	1093, 3511	Same as base 2, because $8 = 2^3$.
9	11, 1006003	Same as base 3, because $9 = 3^2$.
10	3, 487, 56598313	(A014129)
11	71
12	2693, 11429, 81537269
13	2, 863, 1747591
14	29, 353, 5834244249152913116
16	1093, 3511	Same as base 2, because $16 = 2^4$.

If a prime $p$ is a Wieferich prime in base $a$, it will also be a Wieferich prime in base $a^k$ for any integer $k$. This is why bases 4, 8, and 16 perfectly mirror base 2, and base 9 mirrors base 3. The computational heavy lifting to find these values is immense, relying on optimized algorithms for modular exponentiation to scan through billions of candidate primes.

I was surprised not to find Arthur Wieferich listed in the MacTutor history of mathematicians but then I came across this interesting information about him on Wikipedia:

Arthur Josef Alwin Wieferich (April 27, 1884 – September 15, 1954) was a German mathematician and teacher, remembered for his work on number theory, as exemplified by a type of prime numbers named after him. 

 

He was born in Münster, attended the University of Münster (1903–1909) and then worked as a school teacher and tutor until his retirement in 1949. He married in 1916 and had no children. 

 

Wieferich abandoned his studies after his graduation and did not publish any paper after 1909. His mathematical reputation is founded on five papers he published while a student at Münster.

The first three papers are related to Waring's problem. His fourth paper led to the term Wieferich prime, which are p such that \(p^2\) divides \(2^(p-1) - 1\).

I got Gemini to research why he never published anything after 1909: 

The historical record does not contain a definitive letter, diary entry, or professional correspondence where Arthur Wieferich explicitly states why he walked away from research. Biographical information on him is extremely sparse, largely compiled from the archives of the University of Münster and state records in Hannover. However, piecing together his timeline and the structure of the German academic system at the time provides a clear picture of why a mathematician of his caliber would vanish from the literature.

The German Academic Bottleneck

In the early 20th century, publishing groundbreaking papers—as Wieferich did between \( 1908 \) and \( 1909 \)—was not a guaranteed ticket to a university professorship. To secure a permanent academic post in Germany, a scholar had to complete a Habilitation and then work for years as a Privatdozent. A Privatdozent was an external lecturer who received no salary from the university, relying solely on the meager fees paid by the specific students attending their lectures.

Unless a young mathematician had independent wealth or substantial family backing, this path was financially ruinous. Wieferich was the son of a businessman, but evidently chose the alternative: the highly respected, financially secure civil service path of a Gymnasiallehrer (secondary school teacher).

The Trajectory of a Teacher

Wieferich's mathematical burst occurred entirely while he was a student, likely heavily influenced by a number theory course taught by Max Dehn in \( 1907 \). His famous paper on Fermat's Last Theorem, which defined what are now called Wieferich primes—primes \( p \) such that \( p^2 \mid 2^{p-1} - 1 \)—was published when he was just \( 25 \).

Upon graduating in \( 1909 \), he passed the state examination and immediately entered the school system. He was not a researcher who slowly faded away due to lack of ideas; he simply changed careers upon graduation. He taught in a succession of towns: Konitz, Elbing, Zoppot, Neustadt, Jülich, Stade, and finally Meppen.

He did not sever ties with the mathematical community immediately. He remained a member of the Deutsche Mathematiker-Vereinigung (the German Mathematical Society) until about \( 1929 \), twenty years after his last publication. This suggests he maintained a personal interest in the field, even if his profession left no time or incentive for active research.

A Minor Footnote: The Kempner Correction

There is one mathematical footnote regarding his departure. His first paper in \( 1908 \) tackled Waring's problem, attempting to prove that every integer can be represented as the sum of at most \( 9 \) positive cubes. The formidable Edmund Landau publicly praised the paper as a massive advance in elementary number theory.

However, Wieferich's proof contained a logical gap. It was not fatal to the overall idea, but it required an external correction, which was published by Aubrey J. Kempner in \( 1912 \). There is no historical evidence suggesting this correction discouraged Wieferich—he was already three years into his teaching career by the time Kempner published the fix—but it is the only major post-publication event related to his work during his lifetime.

Following World War II, from \( 1945 \) until his retirement in \( 1949 \), Wieferich supported himself as a private tutor. He died in \( 1954 \) in Meppen, leaving behind no children and only five papers—yet enough of a legacy to ensure his name remains permanently attached to number theory.