On prime values of cyclotomic polynomials

Mathematics

Finding irreducible polynomials over Q (or over Z ) is not always easy. However, it is well-known that the mth cyclotomic polynomials are irreducible over Q . In this paper, we define the mth modified cyclotomic polynomials and we get more irreducible polynomials over Q systematically by using the modified cyclotomic polynomials. Since not all modified cyclotomic polynomials are irreducible, a criterion to decide the irreducibility of those polynomials is studied. Also, we count the number of irreducible mth modified cyclotomic polynomials when m = p α with p a prime number and α a positive integer.

Acta Arithmetica, 1999

Mathematika, 2021

Given any positive integer n, let A(n) denote the height of the n th cyclotomic polynomial, that is its maximum coefficient in absolute value. It is well known that A(n) is unbounded. We conjecture that every natural number can arise as value of A(n) and prove this assuming that for every pair of consecutive primes p and p ′ with p ≥ 127 we have p ′ −p < √ p+1. We also conjecture that every natural number occurs as the maximum coefficient of some cyclotomic polynomial and show that this is true if Andrica's conjecture holds, i.e., that √ p ′ − √ p < 1 always holds. This is the first time, as far as the authors know, that a connection between prime gaps and cyclotomic polynomials is uncovered. Using a result of Heath-Brown on prime gaps we show unconditionally that every natural number m ≤ x occurs as A(n) value with at most O ǫ (x 3/5+ǫ) exceptions. On the Lindelöf Hypothesis we show there are at most O ǫ (x 1/2+ǫ) exceptions and study them further by using deep work of Bombieri-Friedlander-Iwaniec on the distribution of primes in arithmetic progressions beyond the square-root barrier.

The Electronic Journal of Combinatorics, 2012

We pose the question of determining the lowest-degree polynomial with nonnegative coefficients divisible by the $n$-th cyclotomic polynomial $\Phi_n(x)$. We show this polynomial is $1 + x^{n/p} + \cdots + x^{(p-1)n/p}$ where $p$ is the smallest prime dividing $n$ whenever $2/p &gt; 1/q_1 + \cdots + 1/q_k$, where $q_1, \ldots, q_k$ are the other (distinct) primes besides $p$ dividing $n$. Determining the lowest-degree polynomial with nonnegative coefficients divisible by $\Phi_n(x)$ remains open in the general case, though we conjecture the existence of values of $n$ for which this degree is, in fact, less than $(p-1)n/p$.

Primitive prime divisors play an important role in group theory and number theory. We study a certain number theoretic quantity, called Φ * n (q), which is closely related to the cyclotomic polynomial Φ n (x) and to primitive prime divisors of q n − 1. Our definition of Φ * n (q) is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants c and k, we give an algorithm for determining all pairs (n, q) with Φ * n (q) cn k. This algorithm is used to extend (and correct) a result of Hering which is useful for classifying certain families of subgroups of finite linear groups.

2008

Let Φ n (x) denote the nth cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that a n (k), the coefficient of x k in Φ n (x), satisfies |a n (k)| ≤ (p + 1)/2 in case n = pqr with p < q < r primes (in this case Φ n (x) is said to be ternary). Since then several results towards establishing her conjecture have been proved (for example |a n (k)| ≤ 3p/4). Here we show that, nevertheless, Beiter's conjecture is false for every p ≥ 11. We also prove that given any ǫ > 0 there exist infinitely many triples (p j , q j , r j ) with p 1 < p 2 < . . . consecutive primes such that |a p j q j r j (n j )| > (2/3ǫ)p j for j ≥ 1.

The American Mathematical Monthly, 1980

This note shows how some basic techniques of algebraic number theory can be used to answer completely some questions recently raised in the Problems section of this MONTHLY [6]. We consider the sequence N PN=PN(f)= II f(rl.) (1) j=l where f(x) is a fixed polynomial with integer coefficients and tN = exp(27Ti / N). Since PN is an algebraic integer in Q(tN) and invariant under all the automorphisms tN~t;., (k,N)= I of that field, PN is actually a rational integer. We are interested in the divisibility properties of the sequence PN. This problem was first examined by Pierce [3] and later treated exhaustively by D. H. Lehmer [2). Using Lehmer's methods, we describe a finite computation which determines for a given prime power pk the set of N for whichpkiPN. We apply these methods to the special case f(x)=x 4 +x+ I to prove generalized and corrected forms of conjectures made by P. Bruckman in [ 6). We first give an alternative expression for PN. Letting

Journal of the Australian Mathematical Society, 2015

Primitive prime divisors play an important role in group theory and number theory. We study a certain number-theoretic quantity, called${\rm\Phi}_{n}^{\ast }(q)$, which is closely related to the cyclotomic polynomial${\rm\Phi}_{n}(x)$and to primitive prime divisors of$q^{n}-1$. Our definition of${\rm\Phi}_{n}^{\ast }(q)$is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants$c$and$k$, we provide an algorithm for determining all pairs$(n,q)$with${\rm\Phi}_{n}^{\ast }(q)\leq cn^{k}$. This algorithm is used to extend (and correct) a result of Hering and is useful for classifying certain families of subgroups of finite linear groups.

Involve, a Journal of Mathematics, 2011

A cyclotomic polynomial Φ n (x) is said to be ternary if n = pqr with p, q and r distinct odd primes. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Here we establish some results and formulate some conjectures regarding the coefficients appearing in the polynomial family Φ pqr (x) with p < q < r, p and q fixed and r a free prime.

2011

A cyclotomic polynomial Φ n (x) is said to be ternary if n = pqr with p, q and r distinct odd primes. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Here we establish some results and formulate some conjectures regarding the coefficients appearing in the polynomial family Φ pqr (x) with p < q < r, p and q fixed and r a free prime.