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LoginProving that certain non-linear diophantine equations have infinitely many (or no) solutions
Course Queries Syllabus Queries 3 years ago
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manpreet![Tuteehub forum best answer]()
Best Answer
3 years ago
How do I find the solubility of certain non-linear diophantine equations:
For eg.: x4+y4=z4x4+y4=z4 is insoluble in NN, which is easy to prove by infinite descent, but x2+y2=z3x2+y2=z3has infinitely many solutions (which I am apparently stuck with).
The text from where this problem came (Elementary Number Theory, David M. Burton) contains a hint stating to choose x=n(n2−3)x=n(n2−3) and y=3n2−1y=3n2−1 which trivializes the problem. I want to know how to understand the reasoning behind the choice of such substitutions.
Also, as a soft question, is it alright to simultaneously ready An Introduction to the Theory of Numbers by G.H. Hardy along with my present syllabus, or should I focus more on one text?