quadripple equation

[Tahmid Hasan]

find all natural number solutions of this equation
$a^4+b^4=2007^{2007}$

[Moon]

$\pmod{4}$

[Tahmid Hasan]

moon bhai .don't understand ur hint

[Moon]

$a^4+b^4 \equiv 0,1,2 \pmod{4}$, but $2007^{2007} \equiv (-1)^{odd} \equiv 3 \pmod{4}$.

[Tahmid Hasan]

yes there is no natural number sol,i did it with the digit of the powers

[Masum]

yes there is no natural number sol,i did it with the digit of the powers

could you tell me how that is?

[tanmoy]

Moon via,why have you taken $mod 4$???

[Labib]

Moon via,why have you taken $mod 4$???

He took $a^4+b^4$ which is congruent to $0,1$ or $2$ (mod $4$) which kills the problem.

[Phlembac Adib Hasan]

Moon via,why have you taken $mod 4$???

Did you want to know how Moon vai realized mod 4 would kill this problem?
That's intuition. Taking $\bmod\; 3,4,6,7,9$ is always very helpful. Sometimes the problem itself provides some valuable information. For example, here we had to deal with some fourth powers. So $\bmod \;3,4,8$ are the smartest choices, since $x^2 \equiv 0,1,4 \pmod {3,4,8}$. If there are some cubes, we may consider $\bmod\; 7,9$.