# Gaps in sequantial list.

1. Feb 20, 2012

### 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?

2. Feb 20, 2012

### 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.

Last edited: Feb 20, 2012
3. Feb 20, 2012

### Stephen Tashi

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.