## The family of ternary cyclotomic polynomials with one free prime.(English)Zbl 1292.11052

Ternary cyclotomic polynomials are cyclotomic polynomials $$\Phi_{pqr}$$, where $$p$$, $$q$$ and $$r$$ are distinct odd primes, $$3\leq p<q<r$$ say. Following the standard convention, we write $$A(pqr)$$ for the greatest absolute value of the coefficients of $$\Phi_{pqr}$$, its height. This function is probably the most studied question about the coefficients of these polynomials, but a number of interesting and challenging problems remain open. One also writes $$M(p)=\max_{q,r}A(pqr)$$ and $$M(p,q)=\max_{r}A(pqr)$$. The paper under review considers certain functional properties of $$M(p,q)$$. It helps to recall the following result due to N. Kaplan [J. Number Theory 127, No. 1, 118–126 (2007; Zbl 1171.11015)]. For fixed $$p$$ and $$q$$, the value of $$A(pqr)$$ depends only on the residue class of $$r$$ to the modulus $$pq$$. It is only natural to consider whether, for a fixed $$p$$, $$M(p,q)$$ is determined by $$q\bmod p$$. This is false, as is shown by the example $$M(7,13)=3$$ and $$M(7,41)=M(7)=4$$. Nonetheless, the authors conjecture that the appropriate analogue of Kaplan’s result for the function $$M(p,q)$$ does exist.
Let us present their conjecture in the form of two questions: Is it true that for (i) $$q>p^3$$ the value of $$M(p,q)$$ depends only on (ii) the residue class of $$q$$ to the modulus $$p^3$$? What are the correct quantities that should be used in place of $$p^3$$ in (i) and in (ii)?
The conjecture is supported by the following result. For a fixed $$p$$ and an arbitrary integer $$a$$, there are integers $$q_a$$ and $$d_a$$ such that the value of $$M(p,q)$$ remains unchanged for all $$q\geq q_a$$ with $$q\equiv a\pmod{d_ap}$$. Observing an application of this result to the equation $$M(p,q)=M(p)$$, the authors conclude that the set of primes $$q$$ satisfying this equation always contains an arithmetic progression and thus has a positive lower asymptotic density in the set of primes. For certain small primes $$p$$ they were able to obtain a more complete analysis of this problem, lending further support for their conjecture. We give their results for $$3, 5$$ and $$7$$, which are complete and particularly simple, if somewhat confusing. For $$p=3$$ or $$5$$, we have $$M(p,q)\equiv M(p)$$. That is, in these cases $$M(p,q)$$ is determined by $$q\bmod p$$, but happens to be constant. Similarly, $$M(7,q)=M(7)$$ for all $$q\neq13$$, and we again observe periodicity modulo $$p$$ for all $$q\geq17$$.
In addition, the paper contains a number of other results on $$M(p,q)$$ – too numerous to mention here – and a collection of open questions.

### MSC:

 11C08 Polynomials in number theory 11B83 Special sequences and polynomials

### Keywords:

ternary cyclotomic polynomial; coefficient

Zbl 1171.11015
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