Polynomials evaluated at integers by John D. Cook
Let p(x) = a0 + a1x + a2x2 + … + anxn and suppose at least one of the coefficients ai is irrational for some i ≥ 1. Then a theorem by
Weyl says that the fractional parts of p(n) are equidistributed as n varies over the integers. That is, the proportion of values that land
in some interval is equal to the length of that interval.
Clearly it’s necessary that one of the coefficients be irrational. What may be surprising is that it is sufficient.
If the coefficients are all rational with common denominator N, then the sequence would only contain multiples of 1/N. The interval
[1/3N, 2/3N], for example, would never get a sample. If a0 were irrational but the rest of the coefficients were rational, we’d have the
same situation, simply shifted by a0.
This is a theorem about what happens in the limit, but we can look at what happens for some large but finite set of terms. And we can
use a χ2 test to see how evenly our sequence is compared to what one would expect from a random sequence.
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