Sum of the Solution

Number Theory Level 3

Find the number of all ordered integer pairs ( x , y ) (x , y) that satisfy x 3 + 117 y 3 = 5 x^3 + 117y^3 = 5 .


The answer is 0.

Alan Yan by Alan Yan , 20, USA

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1 solution

Alan Yan
Sep 9, 2015

In modulo 9, perfect cubes can only be 0,1,-1 mod 9. Taking mod 9 \text{mod 9} and noticing 117 is 0 in modulo 9 , it is clear that there are no solutions.

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It's better to explain why you chose mod 9.

Pi Han Goh - 5 years, 9 months ago

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edited for the sake of the community

Alan Yan - 5 years, 9 months ago

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Nah. I was going for "because 117 is divisible by 9, we can simplify it to modulo 9, ...."

Pi Han Goh - 5 years, 9 months ago

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@Pi Han Goh – I think knowing that cubes must be 1, 0 , -1 is more important than that trivial fact... but okay.

Alan Yan - 5 years, 9 months ago

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