1. The odd complete square number mod 8 is 1, so the odd fourth power is also mod 8 is 1.

2. Even powers can be divisible by 16, so mod 8 is greater than 0.

So the sum of the four powers of seven integers mod 8 is the same as the odd number of these seven numbers.

And 5000 ≡ 0 (mod 8), so if the sum of the four powers of seven integers is equal to 5000, then they must all be even numbers.

But even power can be divisible by 16, and even power can also be divisible by 16.

The result is that 5000 is divisible by 16, which is contradictory.

So there is no sum of the four powers of seven integers equal to 5000, that is, the equation has no integer solution.