1,713 research outputs found

    Diophantine equations with Euler polynomials

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    In this paper we determine possible decompositions of Euler polynomials Ek(x)E_k(x), i.e. possible ways of writing Euler polynomials as a functional composition of polynomials of lower degree. Using this result together with the well-known criterion of Bilu and Tichy, we prove that the Diophantine equation 1k+2k+(1)xxk=g(y),-1^k +2 ^k - \cdots + (-1)^{x} x^k=g(y), with gQ[X]g\in \mathbb{Q}[X] of degree at least 22 and k7k\geq 7, has only finitely many integers solutions x,yx, y unless polynomial gg can be decomposed in ways that we list explicitly.Comment: to appear in Acta Arithmetic

    Decomposable polynomials in second order linear recurrence sequences

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    We study elements of second order linear recurrence sequences (Gn)n=0(G_n)_{n= 0}^{\infty} of polynomials in C[x]\mathbb{C}[x] which are decomposable, i.e. representable as Gn=ghG_n=g\circ h for some g,hC[x]g, h\in \mathbb{C}[x] satisfying degg,degh>1\operatorname{deg}g,\operatorname{deg}h>1. Under certain assumptions, and provided that hh is not of particular type, we show that degg\operatorname{deg}g may be bounded by a constant independent of nn, depending only on the sequence.Comment: 26 page

    On equal values of power sums of arithmetic progressions

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    In this paper we consider the Diophantine equation \begin{align*}b^k +\left(a+b\right)^k &+ \cdots + \left(a\left(x-1\right) + b\right)^k=\\ &=d^l + \left(c+d\right)^l + \cdots + \left(c\left(y-1\right) + d\right)^l, \end{align*} where a,b,c,d,k,la,b,c,d,k,l are given integers. We prove that, under some reasonable assumptions, this equation has only finitely many integer solutions.Comment: This version differs slightly from the published version in its expositio

    Will physical activity increase academic performance?

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    The purpose of this synthesis project is to collectively assess and analyze the critical mass of research articles to determine if physical activity can increase academic performance. The studies that met the inclusion criteria of examining the effects of physical activity on academic performance were included in this project. The articles were analyzed by using a synthesis grid, which helped to organize and examine the methods, results and discussions. Themes, such as effects on executive functioning, effects on academic core subjects and weekly hours, were accumulated from the articles. Based upon the critical mass; the results indicate that there can be a positive relationship between physical activity and academic performance. From this, discussion points were concluded examining grade level differences, advocating for physical education, comprehensive school physical activity program, after school sport participation, limitations and recommendations for future research

    On conjectures and problems of Ruzsa concerning difference graphs of S-units

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    Given a finite nonempty set of primes S, we build a graph G\mathcal{G} with vertex set Q\mathbb{Q} by connecting x and y if the prime divisors of both the numerator and denominator of x-y are from S. In this paper we resolve two conjectures posed by Ruzsa concerning the possible sizes of induced nondegenerate cycles of G\mathcal{G}, and also a problem of Ruzsa concerning the existence of subgraphs of G\mathcal{G} which are not induced subgraphs.Comment: 15 page