Randomly choose a subset $\{a,b,c,d\}$ of size exactly 4 uniformly from among all $n \choose 4$ size 4 subsets of $\{1,2,3,...,N\}$.

Form the polynomial $p(z) = 1 + z^a + z^b + z^c + z^d$.

What's the probability that $p(z)$ has at least one root on the unit circle? The probability depends on $N$, but how does the probability behave when $N$ approaches infinity?

Some plausible conjectures imply that the answer is $909/9464$. See paper A6 here: http://math.colgate.edu/~integers/vol12.html

Randomly choose a subset $\{a,b,c,d\}$ of size exactly 4 uniformly from among all $N \choose 4$ size 4 subsets of $\{1,2,3,...,N\}$.

Form the polynomial $p(z) = 1 + z^a + z^b + z^c + z^d$.

What's the probability that $p(z)$ has at least one root on the unit circle? The probability depends on $N$, but how does the probability behave when $N$ approaches infinity?

Some plausible conjectures imply that the answer is $909/9464$. See paper A6 here: http://math.colgate.edu/~integers/vol12.html