کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
5772705 | 1630640 | 2017 | 13 صفحه PDF | دانلود رایگان |
عنوان انگلیسی مقاله ISI
On certain Diophantine equations of the form z2 = f(x)2 ± g(y)2
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کلمات کلیدی
موضوعات مرتبط
مهندسی و علوم پایه
ریاضیات
اعداد جبر و تئوری
پیش نمایش صفحه اول مقاله
چکیده انگلیسی
In this note we study the existence of integer solutions of the Diophantine equationz2=f(x)2±g(y)2 for certain polynomials f,gâZ[x] of degree â¥3. In particular, for given kâN we prove that for all a,bâZ satisfying the condition a2+b2â 0, the above Diophantine equation, with f(x)=xk(x+a), g(x)=xk(x+b) and any choice of the sign, has infinitely many solutions in integers x,y,z such that f(x)g(y)â 0. Moreover, we prove that for f(x)=x3 and 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,z2 with gcdâ¡(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)2 has infinitely many solutions in integers.
ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Number Theory - Volume 174, May 2017, Pages 239-257
Journal: Journal of Number Theory - Volume 174, May 2017, Pages 239-257
نویسندگان
Sz. Tengely, M. Ulas,