## The Fine-Wilf Theorem

The Fine-Wilf Theorem is as follows. Supppose and are two infinite periodic sequences with periods and respectively. If and agree on their first terms then the two sequences are identical.

We present here the elegant algebraic proof given as the solution to Problem 134 in Bollobas’ book *The Art of Mathematics* and attributed by Bollobas to Ernst Straus.

We first define formal power series and . By the periodicity hypothesis, we have and for some polynomials of maximum degrees and respectively. However, since divides each of and , we also have

where is some polynomial of degree at most . If the first coefficients of and coincide, then the first coefficients of are zero, whence the polynomial is the zero polynomial. It follows that is identically zero, so that and and hence and are identical, as claimed.

We should point out that several other results that we have discussed in previous posts appear as problems in the previously mentioned book by Bollobas.

- Cauchy-Davenport is Problem 96.
- Erdos-Ginzburg-Ziv is Problem 97.
- Bollobas’ Cross-Intersecting Pairs Theorem is a weaker version of a result of Frankl given as Problem 113.
- Problem 115 is a consequence of the two families result discussed here.

## Leave a Reply