Related papers: Cyclotomic primes

In this article we study the irreducibility of polynomials of the form $x^n+\epsilon_1 x^m+p^k\epsilon_2$, $p$ being a prime number. We will show that they are irreducible for $m=1$. We have also provided the cyclotomic factors and…

The problem of simplicity of Fermat number-twins $f_{n}^{\pm}=2^{2^n}\pm3$ is studied. The question for what $n$ numbers $f_{n}^{\pm}$ are composite is investigated. The factor-identities for numbers of a kind $x^2 \pm k $ are found.

We give all non splitting bi-unitary perfect polynomials over the prime field of two elements, which have only Mersenne polynomials as odd irreducible divisors.

We explore the connection between cyclotomic mapping permutation polynomials and permutation polynomials of the form $x^rf(x^{\frac{q-1}{l}})$ over finite fields. We present a new necessary and a new sufficient condition to verify…

Prime number related fractal polygons and curves are derived by combining two different aspects. One is an approximation of the prime counting function build on an additive function. The other are prime number indexed basis entities taken…

We present some new ideas on important problems related to primes. The topics of our discussion are: simple formulae for primes, twin primes, Sophie Germain primes, prime tuples less than or equal to a predefined number, and their…

The two Fresnel Integrals are real and imaginary part of the integral over complex-valued exp(ix^2) as a function of the upper limit. They are special cases of the integrals over x^m*exp(i*x^n) for integer powers m and n, which are…

This paper studies the largest cycles consisted by the quadratic residues modulo prime numbers. We give some formulae about the maximum length of the cycles. Especially, the formula for modulo Fermat primes is given.

We generalize the definition and properties of root systems to complex reflection groups - roots become rank one projective modules over the ring of integers of a number field k. In the irreducible case, we provide a classification of root…

It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of…

This paper investigates coefficients of cyclotomic polynomials theoretically and experimentally. We prove the following result. {{\em If $n=p_1\ldots p_k$ where $p_i$ are odd primes and $p_1<p_2<\ldots<p_r<p_1+p_2<p_{r+1}<\ldots<p_t$ with…

A unitary cyclotomic polynomial of order three is a polynomial of the form \[ \Phi^*_{PQR}(x)=\frac{(x^{PQR}-1)(x^P-1)(x^Q-1)(x^R-1)}{(x^{PQ}-1)(x^{QR}-1)(x^{RP}-1)(x-1)}, \] where $P$, $Q$ and $R$ are powers of three distinct primes $p$,…

The $n^{th}$ cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Its coefficients are the subject of intensive study and some formulas are known for them. Here we are interested in formulas…

Let $n$ be a product of two distinct prime numbers. We construct a triangulated monoidal category having a Grothendieck ring isomorphic to the ring of $n$:th cyclotomic integers.

We continue to explore cyclotomic factors in the descent set polynomial $Q_{n}(t)$, which was introduced by Chebikin, Ehrenborg, Pylyavskyy and Readdy. We obtain large classes of factors of the form $\Phi_{2s}$ or $\Phi_{4s}$ where $s$ is…

In this paper we generalize the result of Fouvry and Iwaniec dealing with prime values of the quadratic form $x^2 + y^2$ with one input restricted to a thin subset of the integers. We prove the same result with an arbitrary primitive…

This note investigates the prime values of the polynomial $f(t)=qt^2+a$ for any fixed pair of relatively prime integers $ a\geq 1$ and $ q\geq 1$ of opposite parity. For a large number $x\geq1$, an asymptotic result of the form $\sum_{n\leq…

Let $\Psi_n(x)$ be the monic polynomial having precisely all non-primitive $n$th roots of unity as its simple zeros. One has $\Psi_n(x)=(x^n-1)/\Phi_n(x)$, with $\Phi_n(x)$ the $n$th cyclotomic polynomial. The coefficients of $\Psi_n(x)$…

We investigate the solvability of the Diophantine equation $x^2-my^2=\pm p$ in integers for certain integer $m$ and prime $p$. Then we apply these results to produce family of maximal real subfield of a cyclotomic field whose class number…

Let $m$ and $n$ be positive integers, and let $p$ be a prime. Let $T(x)=\Phi_{p^m}\left(\Phi_{2^n}(x)\right)$, where $\Phi_k(x)$ is the cyclotomic polynomial of index $k$. In this article, we prove that $T(x)$ is irreducible over $\mathbb…