کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4593685 | 1630665 | 2015 | 13 صفحه PDF | دانلود رایگان |
In this paper, we consider some hybrid Diophantine equations of addition and multiplication. We first improve a result on new Hilbert–Waring problem. Then we consider the equationequation(1){A+B=CABC=Dn where A,B,C,D,n∈Z+A,B,C,D,n∈Z+ and n≥3n≥3, which may be regarded as a generalization of Fermat's equation xn+yn=znxn+yn=zn. When gcd(A,B,C)=1gcd(A,B,C)=1, (1) is equivalent to Fermat's equation, which means it has no positive integer solutions. We discuss several cases for gcd(A,B,C)=pkgcd(A,B,C)=pk where p is an odd prime. In particular, for k=1k=1 we prove that (1) has no nonzero integer solutions when n=3n=3 and we conjecture that it is also true for any prime n>3n>3. Finally, we consider Eq. (1) in quadratic fields Q(t) for n=3n=3.
Journal: Journal of Number Theory - Volume 149, April 2015, Pages 33–45