On Wednesday, November 20, 2013 1:50:59 AM UTC-6, Robert Crandal wrote: > However, I'd now > like to revise the data and requirements. > Here is the new data set: [etc.]
> Can this data be represented with a formula > that only uses either addition, subtraction, > multiplication, division, modulus, or the power (^) > function, or any combination of these?
All finite data sets can be represented by polynomials. So that means: only constants, addition, subtraction and multiplication.
If the points are A, B, C, ..., Z, then the unique polynomial of minimal degree that is equal to 1 at A and 0 everywhere else is p_A(x) = ((x-B)/(A-B)) ((x-C)/(A-C)) ... ((x-Z)/(A-Z)). Similar considerations apply to the other points, yielding definitions for p_B(x), p_C(x), ..., p_Z(x).
The unique polynomial of minimal degree that matches the value a at A, b at B, c at C, ..., z at Z is then p(x) = a p_A(x) + b p_B(x) + c p_C(x) + ... + z p_Z(x).
All other polynomials matching these points have p(x) as a factor.
Any other function that matches the points must necessarily involve one of the other operations: division, modulus, exponentiation, or something else besides those.