Squarefree values of trinomial discriminants

  title={Squarefree values of trinomial discriminants},
  author={David W. Boyd and Greg Martin and Mark Thom},
  journal={Lms Journal of Computation and Mathematics},
The discriminant of a trinomial of the form x n x m 1 has the form n n (n m) n m m m if n and m are relatively prime. We investigate when these discriminants have nontrivial square factors. We explain various unlikely-seeming parametric families of square factors of these discriminant values: for example, whenn is congruent to 2 (mod 6) we have that ((n 2 n+1)=3) 2 always divides n n (n 1) n 1 . In addition, we discover many other square factors of these discriminants that do not t into these… 

Tables from this paper

Cubefree Trinomial Discriminants

The discriminant of a polynomial of the form $\pm x^n \pm x^m \pm 1$ has the form $n^n \pm m^m(n-m)^{n-m}$ when $n,m$ are relatively prime. We investigate when these discriminants have prime power

Infinite families of monogenic trinomials and their Galois groups

Let n ∈ ℤ with n ≥ 3. Let Sn and An denote, respectively, the symmetric group and alternating group on n letters. Let m be an indeterminate, and define fm(x) := xn + a(m,n)x + b(m,n), where

Splitting fields of $X^n-X-1$ (particularly for $n=5$), prime decomposition and modular forms

A BSTRACT . We study the splitting fields of the family of polynomials f n ( X ) = X n − X − 1 . This family of polynomials has been much studied in the literature and has some remarkable properties.

Squarefree parts of discriminants of trinomials

We obtain a lower bound on the number of distinct squarefree parts of the discriminants nn + (−1)n(n−1)n-1 of trinomials $${X^n - X - 1\in \mathbb{Z}[X]}$$Xn-X-1∈Z[X] for $${1 \leqslant n \leqslant

The Irreducibility and Monogenicity of Power-Compositional Trinomials

. A polynomial f ( x ) ∈ Z [ x ] of degree N is called monogenic if f ( x ) is irreducible over Q and { 1 ,θ,θ 2 ,...,θ N − 1 } is a basis for the ring of integers of Q ( θ ), where f ( θ ) = 0.

Monogenic fields arising from trinomials

We call a polynomial monogenic if a root $\theta$ has the property that $\mathbb{Z}[\theta]$ is the full ring of integers in $\mathbb{Q}(\theta)$. Using the Montes algorithm, we find sufficient

Monogenic trinomials with non-squarefree discriminant

For each integer [Formula: see text], we identify new infinite families of monogenic trinomials [Formula: see text] with non-squarefree discriminant, many of which have small Galois group. Moreover,

Monogenic Cyclotomic compositions

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

Survey on irreducibility of trinomials

Let $a,b,c$ be non-zero integers and $f(x)=ax^n+bx^m+c$ be a trinomial of degree $n$. We surveyed the irreducibility criteria of $f(x)$ over rational numbers.

Twisted-PHS: Using the Product Formula to Solve Approx-SVP in Ideal Lattices

It is proved that the Twisted-PHS algorithm performs at least as well as the original PHS algorithm, and the full implementation of this algorithm is provided, which suggests that much better approximation factors are achieved, and that the given lattice bases are a lot more orthogonal than the ones used in PHS.



A construction of polynomials with squarefree discriminants

For any integer n >= 2 and any nonnegative integers r,s with r+2s = n, we give an unconditional construction of infinitely many monic irreducible polynomials of degree n with integer coefficients

An estimate for Heilbronn's exponential sum

for any integer a coprime to p. It is important to note here that if n ≡ n′ (mod p), then n ≡ n′p (mod p). Thus the summand in S(a) has period p with respect to n, so that S(a) is a ‘complete sum’ to

Factorization of polynomials over finite fields.

Dickson [1, Ch. V, Th. 38] has given an interesting necessary condition for a polynomial over a finite field of odd characteristic to be irreducible. In Theorem 1 below, I will give a generalization

On Cauchy–Liouville–Mirimanoff Polynomials

  • P. Tzermias
  • Mathematics
    Canadian Mathematical Bulletin
  • 2007
Abstract Let $p$ be a prime greater than or equal to 17 and congruent to 2 modulo 3. We use results of Beukers and Helou on Cauchy–Liouville–Mirimanoff polynomials to show that the intersection of

The new book of prime number records

1 How Many Prime Numbers Are There?.- I. Euclid's Proof.- II. Goldbach Did It Too!.- III. Euler's Proof.- IV. Thue's Proof.- V. Three Forgotten Proofs.- A. Perott's Proof.- B. Auric's Proof.- C.

Old and new conjectured diophantine inequalities

The original meaning of diophantine problems is to find all solutions of equations in integers or rational numbers, and to give a bound for these solutions. One may expand the domain of coefficients

Wieferich's criterion and the abc-conjecture

13 lectures on Fermat's last theorem

Lecture I The Early History of Fermat's Last Theorem.- 1 The Problem.- 2 Early Attempts.- 3 Kummer's Monumental Theorem.- 4 Regular Primes.- 5 Kummer's Work on Irregular Prime Exponents.- 6 Other

On the Oesterlé-Masser conjecture

Letx, y andz be positive integers such thatx=y+z and ged (x,y,z)=1. We give upper and lower bounds forx in terms of the greatest squarefree divisor ofx y z.