On certain Diophantine equations of the form z² = f(x)² ± g(y)²

详细信息    查看全文

• 作者：Sz. Tengely^a ; ^{tengely@science.unideb.hu} ; M. Ulas^b ; ^{Maciej.Ulas@im.uj.edu.pl}
• 关键词：11D41
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：May 2017
• 年：2017
• 卷：174
• 期：Complete
• 页码：239-257
• 全文大小：382 K
• 卷排序：174

文摘

In this note we study the existence of integer solutions of the Diophantine equationz2=f(x)2±g(y)2z2=f(x)2±g(y)2 for certain polynomials f,g∈Z[x]f,g∈Z[x] of degree ≥3. In particular, for given k∈Nk∈N we prove that for all a,b∈Za,b∈Z satisfying the condition a2+b2≠0a2+b2≠0, the above Diophantine equation, with f(x)=xk(x+a)f(x)=xk(x+a), g(x)=xk(x+b)g(x)=xk(x+b) and any choice of the sign, has infinitely many solutions in integers x,y,zx,y,z such that f(x)g(y)≠0f(x)g(y)≠0. Moreover, we prove that for f(x)=x3f(x)=x3 and g(x)=x(x+1)(x+2)g(x)=x(x+1)(x+2) the system of Diophantine equationsz12=f(x)2+g(y)2,z22=f(x)2+g(y+1)2 has infinitely many solutions in positive integers x,y,z1,z2x,y,z1,z2 with gcd⁡(x,y)=1gcd⁡(x,y)=1. Similar result is proved for the systemz12=f(x)2+g(y)2,z22=f(x+1)2+g(y)2. We also present some experimental results concerning the construction of polynomials with rational coefficients for which the Diophantine equation z2=f(x)2±g(y)2z2=f(x)2±g(y)2 has infinitely many solutions in integers.