A Gleeful Algorithm
Date:
Butler Undergraduate Research Conference
Abstract: Let $f(n)$ denote the number of representations a positive integer $n$ has as a sum of consecutive primes. For example, $f(17)=2$ since $17$ is prime and $17=2+3+5+7$, with no other representations. In 1963, Moser showed that the average value of $f(n)$ is $\log 2$ asymptotically. Moser also posed four questions on $f(n)$: (1) Is $f(n)=1$ infinitely often? (2) Is $f(n)=k$ solvable for every $k$? (3) Does the set of integers $n$ for which $f(n)=k$ have positive density? (4) Is the $\limsup f(n)=\infty$? In this talk, we will describe and analyze a couple algorithms that compute the histogram $h(x,k)$, the number of $n \le x$ such that $f(n)=k$. We will also give conjectured answers to Moser’s four questions based on data from our histogram for $x=10^{12}$.
See the slides here (341 KB).