What is the mathematical topic related to finding gaps in a sequential list?

[rtal]

If I have a sequence 1,2,3,.. 1000. I can find a gap by summing up and subtrating from the sum of 1.. 1000 (usually with a known formula like n x (n+1) / 2 but with processing power I can just add 1 ... 1000 with a computer program).
If there is a two number gap, I can add squares as well and so have two equations with two variables
SumOfOneTo1000 - SumOfListWithGaps = x + y --- Equation 1
SumOfOneSquareTo1000Sequare - SumOfSquaresFromListWithGaps = sqr(x) + sqr(y) ---- Eq 2
Now I have two equations and two unknown and I can simplify that into a quadratic equation with two roots. The roots are x and y.
So I can a 2 number gap as well.
How far can I go with this logic meaning with cubes and 3 gaps etc.
What category does this problem fall under, is it information theory?
thanks for your help.

 

[ramsey2879]

You are correct, that you can form n equations of n unknowns of the form:

A_(1)^(i) + A_(2)^(i) + ... A_(n)^(i) = X_(i) i = {1,2,...,n}.

But equations with i > 3 would fall in the category of higher algebra and would be difficult to solve.

 

[Stephen Tashi]

would fall in the category of higher algebra

Yes, but since this is the Number Theory section, we should keep in mind that the solutions to these particular equations are known to be integers. The equations can be treated as Diophantine equations.

"Diophantine Equations" is the relevant mathematical topic, not "Information Theory". Information Theory takes place in a setting where there are probability distributions.