# An elementary proof that, for every fixed $n \in \mathbf N^+$, there are infinitely many primes $\equiv -1 \bmod n$

This morning, I made a comment to a comment to a question of Ayman Moussa, only to point out that, among many others, there is an elementary proof of Dirichlet's theorem on the existence of infinitely many primes $p \equiv -1 \bmod n$ for every fixed $n \in \mathbf N^+$. I remember of having read this proof in a paper (of about 10 pages?) concerning some properties of Chebyshev polynomials (of the first kind?). I may be wrong, but I seem also to remember that (i) the paper was relatively old, dating back to the first half of the past century, and (ii) the paper was not really focused on Dirichlet's theorem, but derived the result on the primes as a byproduct. Unfortunately, I couldn't backtrack to the paper, yet. So my question is:

Q. Can anyone here retrieve the "lost reference" alluded to in the above?

I'm conscious that all this sounds more like a riddle than a well-grounded mathematical question, but I don't have more clues to offer than the ones I've already supplied.

Edit 1. Of course, there are many elementary proofs of various cases of Dirichlet's theorem in the literature. Some of them, including I. Schur's proof of the existence of infinitely many primes $p \equiv b \bmod a$ for all $a,b \in \mathbf N^+$ such that $b^2 \equiv 1 \bmod a$, are surveyed in:

S. Gueron and R. Tessler, Infinitely Many Primes in Arithmetic Progressions: The Cyclotomic Polynomial Method, Math. Gaz. 86 (Mar., 2002), No. 505, 110-114.

Schur's theorem (edited after a comment of Emil Jeřábek) covers the special case this thread is focused on, but I'm sure it's not the result I've somewhere in mind (which applies only to the case $b = -1$, in the above notations).

Edit 2. I've just found a 2002 preprint of R. Meštrović that provides an extensive, thorough survey on Dirichlet's theorem: The survey includes a section entirely devoted to elementary proofs of special cases of the theorem (Sect. 3.2). Let me quote from p. 32, where IP is the property of a set of containing infinitely many primes (I will replace the reference numbers in the text with the corresponding items in the bibliography, and fix a typo in the name of H. Hasse, which appears in the current version of Meštrović's manuscript as "M. Hasse", as noted by Todd Leason in the comments):

A short but not quite elementary proof of IP in the progressions $-1 \pmod k$ for each $k \ge 2$ was given by M. Bauer [Über die arithmetische Reihe, J. Reine Angew. Math. 131 (1906), 265-267; transl. of Math. és Phys. Lapok 14 (1905), 313. in 1905/6]. In 1951 T. Nagell [pp. 170-173 in Introduction to Number Theory, Wiley, New York, 1951] gives an elementary proof of IP in arithmetic progression $-1 \pmod k$ with $k \ge 2$. Applying a similar argument to those of Niven and Powell for IP in the progressions $\equiv 1 \pmod k$, in 1950 by H. Hasse [Vorlesungen über Zahlentheorie, 2nd edition, Springer-Verlag, New York, 1964] proved IP in the progressions $-1 \pmod k$ for each $k \ge 2$.

Since I can't yet read German, there are (at least) two cases: Either the "lost reference" is missing from Meštrović's survey, or I am wrong and the paper where I first read of this mysterious proof wasn't really a paper, but Nagell's book (I'll check tomorrow in the library). The plot thickens!

Edit 3. I checked Nagell's book, and I maintain that the proof therein (which, by the way, is a little gem) is not the one I read some years ago. As a next attempt, I will follow Gerry Myerson's suggestion in the comments and have a look to Dickson's History (I'll update this post only if I can find something useful to answer this question, or to fix a mistake or whatsoever).

• "Despicably"? I wouldn't be so hard on myself... "Regrettably" perhaps? (Never mind - I see an edit was made already.) Apr 27, 2017 at 15:36
• @JarekKuben So the proof I was attributing to R. Murty is much older (and goes back to I. Schur), with Murty having showed that, in a way, the method cannot be pushed further: I didn't know that, and fixed the OP accordingly. Thanks! --- Edit. Jarek's comment has disappeared, but it pointed to a short write-up of K. Conrad on "Euclidean proofs of Dirichlet's theorem." Apr 27, 2017 at 16:03