Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Find all integer solutions x; y to the following Diophantine equations:



Can anyone please help me get started? I don't have any idea ( Im guessing it will involve mods )

share|improve this question
Getting started...how about graphing software? It's barely legal, but at least we can have some grip as to see if there are solutions (I assume there are) and what they might be. –  imranfat Feb 26 '14 at 16:52

1 Answer 1

We need $$y^3=y^4-77\iff 77=y^4-y^3=y^3(y-1)$$ and $(y,y-1)=1\iff(y^3,y-1)=1$

$y$ being integer and $77=7\cdot11$ contains no cubes, we need $y^3=\pm1$

As commented the Questions are independent, we have $$77=(y^2-x)(y^2+x)$$

Find the integer factors of $77$

For example, $$y^2-x=-1,y^2+x=-77$$

share|improve this answer
These are seperate equations, sorry i didnt make it clear –  John Feb 26 '14 at 16:52
@John, find the edited version –  lab bhattacharjee Feb 26 '14 at 16:56
So we aren't looking for x,y combinations that satisfy both equations at the same time?? –  imranfat Feb 26 '14 at 16:56
No we are not, I dont really understand the bit when we get z^6 –  John Feb 26 '14 at 16:58
@John Ahh I get it. Perhaps it was better to post these seperately. Because I bet more than a dime that people read it as a system. –  imranfat Feb 26 '14 at 17:17

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.