There is a polynomial expression that will match the file. I wonder if expressing it would be compressible to a lower size than original file? [edit] I’m wrong - not all sequences can be expressed with a polynomial.

This reminds me of a data compression scheme I came up with once:

Treat an n bit file as a polynomial over the finite field with characteristic 2. Now, there are some irreducible polynomials in this field, but many polynomials have factors of x and of (x+1). Factor the polynomial into P(x)x^n (x+1)^m. and just collect these terms, storing only P, n, and m.