A Polynomial Diophantine

Number Theory Level 4

How many positive integral solutions are there for the following equation?

2 x 3 + 15 x 2 + 39 x + 35 = y 9 \large 2x^3+15x^2+39x+35=y^9

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The answer is 0.

Shreyansh Mukhopadhyay by Shreyansh Mukhopadhyay , 18, India

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1 solution

Mark Hennings
May 3, 2018

Since the equation is equivalent to ( x + 2 ) 3 + ( x + 3 ) 3 = y 9 (x + 2)^3 + (x + 3)^3 \; = \; y^9 and the equation a 3 + b 3 = c 3 a^3 + b^3 = c^3 is well-known not to have any solutions in positive integers, the answer is 0 \boxed{0} .

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Thank you sir, but I doubt that we can use the Fermat's Last theorum in Math olympiads(although I myself did that). Is there any other method of doing it?

Shreyansh Mukhopadhyay - 3 years, 1 month ago

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Well you could reproduce part of the proof of this part of FLT - it was originally proved by Euler.

Mark Hennings - 3 years, 1 month ago

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