On subsets of GF( q 2 ) with d th power differences

Acta Arithmetica, 2001

Quarterly journal of mathematics, 1998

From Theorem 1 we obtain the following corollaries immediately.

Journal of Number Theory, 2008

Using Faltings' theorem on the Mordell conjecture, we prove that for any prime p 5 and integer n 2, the cyclotomic symbols {a, Φ n (a)} do not form a subgroup of K 2 (Q). This partially confirms a conjecture of Browkin.

ArXiv, 2017

In this note we construct a series of small subsets containing a non-d-th power element in a finite field by applying certain bounds on incomplete character sums. Precisely, let $h=\lfloor q^{\delta}\rfloor>1$ and $d\mid q^h-1$. If $q^h-1$ has a prime divisor $r$ with $r=O((h\log q)^c)$, then there is a constant $0 0$ shows that there exists an explicit subset of cardinality $q^{1-d}=O(\log^{2+\epsilon'}(q^h))$ containing a non-quadratic element in $\mathbb{F}_{q^h}$. On the other hand, the choice of $h=2$ shows that for any odd prime power $q$, there is an explicit subset of cardinality $O(\sqrt{q})$ containing a non-quadratic element in $\mathbb{F}_{q^2}$, essentially improving a $O(q)$ construction by Coulter and Kosick \cite{CK}. In addition, we obtain a similar construction for small sets containing a primitive element. The construction works well provided $\phi(q^h-1)$ is very small, where $\phi$ is the Euler's totient function.

Groups, Difference Sets, and the Monster, 1996

Which groups G contain difference sets with the parameters (v, k, A.)= (q 3 + 2q 2 , q 2 + q, q), where q is a power of a prime p? Constructions of K. Takeuchi, R.L. McFarland, and J.F. Dillon together yield difference sets with these parameters if G contains an elementary abelian group of order q 2 in its center. A result of R.J. Turyn implies that if G is abelian and p is self-conjugate modulo the exponent of G, then a necessary condition for existence is that the exponent of the Sylow p-subgroup of G be at most 2q when p = 2 and at most q if p is an odd prime. In this paper we lower these exponent bounds when q =f. p by showing that a difference set cannot exist for the bounding exponent values of 2q and q. Thus if there exists an abelian (96, 20, 4)-difference set, then the exponent of the Sylow 2-subgroup is at most 4. We also obtain some nonexistence results for a more general family of (v, k, A.)-parameter values.

2013

From some works of P. Furtwängler and H.S. Vandiver, we put the basis of a new cyclotomic approach to Fermats last theorem for p > 3 and to a stronger version called SFLT, by introducing governing fields of the form Q(μq−1) for prime numbers q. We prove for instance that if there exist infinitely many primes q, q 6≡ 1 (mod p), qp−1 6≡ 1 (mod p2), such that for q | q in Q(μq−1), we have q1−c = ap (α) with α ≡ 1 (mod p2) (where c is the complex conjugation), then Fermats last theorem holds for p. More generally, the main purpose of the paper is to show that the existence of nontrivial solutions for SFLT implies some strong constraints on the arithmetic of the fields Q(μq−1). From there, we give sufficient conditions of nonexistence that would require further investigations to lead to a proof of SFLT, and we formulate various conjectures. This text must be considered as a basic tool for future researches (probably of analytic or geometric nature). Résumé. Reprenant des travaux de P....

Rocky Mountain Journal of Mathematics, 2007

It this paper, we study the problem of determining the elements in the rings of integers of quadratic fields Q( √ d) which are representable as a difference of two squares. The complete solution of the problem is obtained for integers d which satisfy conditions given in terms of solvability of certain Pellian equations.

Finite Fields and Their Applications, 1998

Annals of Mathematics

Important note To cite this publication, please use the final published version (if applicable). Please check the document version above.

Finite Fields and Their Applications, 2008

Following Beard, we call a polynomial over a finite field F q perfect if it coincides with the sum of its monic divisors. The study of perfect polynomials was initiated in 1941 by Carlitz's doctoral student Canaday in the case q = 2, who proposed the still unresolved conjecture that every perfect polynomial over F 2 has a root in F 2 . Beard, et al. later proposed the analogous hypothesis for all finite fields. Counterexamples to this general conjecture were found by Link (in the cases q = 11, 17) and Gallardo & Rahavandrainy (in the case q = 4). Here we show that the Beard-O'Connell-West conjecture fails in all cases except possibly when q is prime. When q = p is prime, utilizing a construction of Link we exhibit a counterexample whenever p ≡ 11 or 17 (mod 24). On the basis of a polynomial analog of Schinzel's Hypothesis H, we argue that if there is a single perfect polynomial over the finite field F q with no linear factor, then there are infinitely many. Lastly, we prove without any hypothesis that there are infinitely many perfect polynomials over F 11 with no linear factor.