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?

  • 3
    $\begingroup$ 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. $\endgroup$ – Patricia Hersh Oct 8 '12 at 13:46
  • 1
    $\begingroup$ 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!) $\endgroup$ – Nick Gill Oct 8 '12 at 13:54
  • 1
    $\begingroup$ 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. $\endgroup$ – Bernikov Oct 9 '12 at 9:29
  • $\begingroup$ My bad, I misread your question. $\endgroup$ – Bernikov Oct 9 '12 at 9:35

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.


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.