# Cyclotomic polynomials with coefficients $0,\pm1$

Let me begin with what looks like a joke. According to a Bourbaki member, the following conversation occurred during a meeting dedicated to polishing the but-last version of an Algebra Bourbaki volume:

(a Bourbaki member) Why not state explicitly that the coefficients of cyclotomic polynomials are $0,\pm 1$ ?

(another member) Because it's false.

Here is what I am aware: if $n$ has at most two distinct odd prime factors, then the coefficients of $\Phi_n(X)$ are $0,\pm1$ (Migotti, 1883). In other words, this holds true for $n=2^mp^kq^\ell$, where $p,q$ are primes. On the other hand, it is false for $n=105=3\cdot5\cdot7$, because the coefficient of $X^7$ (or of $X^{41}$ as well) is $-2$.

My question is whether there is a complete characterization of those $n$ for which the coefficients of $\Phi_n(X)$ are $0,\pm1$ ? If not, are there other infinite lists of cyclotomic polynomials with this property?

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You might take a look at the paper "The cyclotomic polynomial topologically" by Gregg Musiker and Vic Reiner available at arxiv.org/abs/1012.1844 and the references therein for some pointers on what is known in this direction. –  Patricia Hersh Oct 8 '12 at 13:46
There's a pretty thorough discussion of this sort of question in this preprint: bprim.org/cyclotomicfieldbook/th.pdf (At least it seemed thorough to me, who knows nothing about the question!) –  Nick Gill Oct 8 '12 at 13:54
It is not a characterization, but let $t$ be any positive integer, and take $p_1< \dots <p_t$ (these are $t$ prime numbers) such that $p_1 \geq 3$ and $p_1+p_2>p_t$. Then the coefficients of $X^p$ and $X^{p−2}$ in $\Phi_{2p_1 \dots p_t}$ are $t-1$ and $t-2$ respectively. This way, one can provide an infinity of cyclotomic polynomials with the asked property. The details can be found in \emph{Polynomials} by Prasolov, there is a section about cylotomic polynomials. –  Bernikov Oct 9 '12 at 9:29

There are other families, but there is by no means a complete characterization known. Even for products of three primes there is no complete answer known. A relevant keyword is flat cyclotomic polynomial. Some results to give a flavor of the problem.

The follwing is due to N. Kaplan (from some years ago, Journal of Number Theory):

Let $p,q,r$ be primes (strictly increasing in size), if $r$ is $1$ or $-1$ modulo $pq$ then the coefficients of the $pqr$ cyclotomic pollynomial are only $0,1, -1$.

He also has some periodicity result that allows constructions: let $n$ be an integer and $s,t$ primes strictly greater $n$, and congruent modulo $n$ then the set of coefficients of the $nt$ and $ns$ cyclotomic polynomial coincide.

In 2010 S. Elder obtained further results in this direction.

Let $p,q,r$ be primes (strictly increasing in size), if $r$ is $2$ or $-2$ modulo $pq$ then the coefficients of the $pqr$ cyclotomic polynomial are only $0,1, -1$ if and only if $q$ is $1$ modulo $p$.

Elder also has some additional results for products of three primes, and also for products of four and five primes; see this presentation of Elder where also the above mentioned results can be found.

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quid is correct; there are no complete characterizations of the known such numbers. In my work, now posted to the arXiv at http://arxiv.org/abs/1207.5811, I found the family he cites as well as the first fully general one for products of four primes, both listed in the abstract.

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