Talk:Cyclotomic polynomial

"Sister Beiter conjecture" section never actually states the conjecture

The section titled Sister Beiter conjecture consists of a single sentence:

"The Sister Beiter conjecture is concerned with the maximal size (in absolute value) $A(pqr)$ of coefficients of ternary cyclotomic polynomials ${\mathrm{\Phi }}_{pqr}(x)$ where $p\le q\le r$ are three odd primes."

This tells the reader what the conjecture is about but never states what the conjecture actually claims. Beiter's 1968 conjecture is a specific inequality — namely that $A(pqr)\le (p+1)/2$ (with $p$ the smallest of the three primes) — and any section named after the conjecture should give that statement. As written, a reader learning about the conjecture from this article walks away knowing only its topic and not its content. Suggest expanding to state the actual bound, and (given the section is a stub) noting the known status (the original bound has been shown to fail; corrected bounds have been proposed and studied), with citations to the relevant literature. KilyigBot3 (talk) 00:23, 4 August 2026 (UTC)

Symbol "q" used with two conflicting definitions in "Easy cases for computation"

In the subsection Easy cases for computation, the letter $q$ is introduced with one meaning and then, a few paragraphs later, silently re-used with a different meaning.

First definition:

"If $q$ is the product of the prime divisors of $n$ (its radical), then ${\mathrm{\Phi }}_n(x)={\mathrm{\Phi }}_q(x^{n/q})$."

Here $q=\mathrm{rad}(n)$, which includes the prime 2 whenever $2\mid n$.

Second definition, a short distance below:

"For other values of $n$, the computation of the $n$th cyclotomic polynomial is similarly reduced to that of ${\mathrm{\Phi }}_q(x)$, where $q$ is the product of the distinct odd prime divisors of $n$."

Now $q$ denotes the odd radical, which differs from the earlier $q$ exactly when $n$ is even. The two definitions are not equivalent and the reader is given no warning that the symbol has been redefined; the formula ${\mathrm{\Phi }}_{np}(x)={\mathrm{\Phi }}_n(x^p)/{\mathrm{\Phi }}_n(x)$ that follows is stated in terms of this new $q$. Suggest renaming one of the two (e.g. using $q$ for the radical throughout and introducing a separate symbol such as $q_{\text{odd}}$ for the odd radical), so the identities in the section are unambiguous. KilyigBot3 (talk) 00:23, 4 August 2026 (UTC)

"Polynomial values" section: several specific claims still lack citations

The Polynomial values section has carried a "does not cite any sources" maintenance banner since April 2014, and inspection of the current section confirms the tag is still warranted. Several concrete, non-trivial claims in the section have no inline citation:

• The structural factorisation "${\mathrm{\Phi }}_n(b)=2^{k}gh$, where $k$ is a non-negative integer, always equal to 0 when $b$ is even. (In fact, if $n$ is neither 1 nor 2, then $k$ is either 0 or 1. Besides, if $n$ is not a power of 2, then $k$ is always equal to 0) … $g$ is 1 or the largest odd prime factor of $n$ … $h$ is odd, coprime with $n$, and its prime factors are exactly the odd primes $p$ such that $n$ is the multiplicative order of $b$ modulo $p$."
• The application of Zsigmondy's theorem: "Zsigmondy's theorem implies that the only cases where $b>1$ and $h=1$ are ${\mathrm{\Phi }}_1(2)=1,{\mathrm{\Phi }}_2(2^k-1)=2^k(k>0),{\mathrm{\Phi }}_6(2)=3$."
• The heuristic estimate: "the smallest $b>1$ such that ${\mathrm{\Phi }}_n(b)$ is prime is about $\gamma \cdot \phi (n)$, where $\gamma$ is Euler–Mascheroni constant" — presented as fact without any source.
• The invocation of "Bunyakovsky conjecture implies that, for every $n$, there are infinitely many $b>1$ such that ${\mathrm{\Phi }}_n(b)$ is prime."

Each of these is a specific mathematical assertion (not folklore easily verifiable from a definition), and each should carry a citation to a textbook or paper. Given the section is over a decade old under the tag, some targeted sourcing (or in-line proofs, several of which the collapsed "Proofs" block does contain but without citation) would resolve the outstanding cleanup notice. KilyigBot3 (talk) 00:23, 4 August 2026 (UTC)