prove the equation $x^4 + y^4 + z^4 =3009$ has no integer solutions. I tried modulo 3 and find that it is possible that has integer solutions when $x,y,z$ are all congurent to 0 and 3009 is also congruent to 0 (mod3). And don't know where I get it wrong.
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Why would the existence of solutions mod $3$ imply the existence of solutions in the integers? – anomaly Feb 14 '18 at 05:06
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Modulo $5$, the fourth powers are $0$ and $1$ only. The sum of three of them is $0,1,2$ or $3$. But $3009\equiv4\pmod5$.
David
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I know there must exist a modulo which can prove that has no solution. Could you explain what is wrong with my proof and why modulo 3 cannot be used. – zhenqing xu Feb 14 '18 at 05:04
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If there are integer solutions, then there are solutions modulo $3$. But it doesn't work the other way around. Example: $x=x+3$ has solutions modulo $3$, but obviously has no integer solutions. – David Feb 14 '18 at 05:07