Find the integral solutions?

Question

Determine all possible integral solutions of the following equation x1^6 + x2^6 + x3^6 + x4^6 + x5^6 + x6^6 + x7^6 = 2699999.
(Here e.g. x1^6 represents x with subscript 1 to the power 6)

Answer 1

Look at the equation mod 7.
2699999 = 1 mod 7.
x^6 = 0 mod 7 if x is 0 mod 7, 1 otherwise.
This implies that exactly one of x1,x2,x3,x4,x5,x6,x7 is not 0 mod 7.
12^6 > 2699999, so the biggest value that xi can have is 11. If we let x1 = 11 and x2 through x7 be 7, then the sum of the left hand side is 2477455.
This is the largest sum that we can get and still be congruent to 1 mod 7 if we keep all the terms less than 2699999. Thus, there are no solutions.

Answer 2

To fatal_flaw_death:

That is an incredibly clever solution!

Dan

Answer 3

-ve infinity to +ve infinity.