Generalizations of Bertrand's Postulate to Sums of Any Number of Primes

Abstract

In 1845, Bertrand conjectured that twice any prime strictly exceeds the next prime. Tchebichef proved Bertrand's postulate in 1850. In 1934, Ishikawa proved a stronger result: the sum of any two consecutive primes strictly exceeds the next prime, except for the only equality $2+3=5$. This observation is a special case of a more general result, perhaps not previously noticed: if $p_n$ denotes the $n$th prime, $n=1, 2, \ldots$, with $p_1=2, p_2=3, \ldots$, and if $c_1, \ldots, c_g$ are nonnegative integers (not necessarily distinct), and $d_1, \ldots, d_h$ are positive integers (not necessarily distinct), and $g>h\ge 1$, then there exists a positive integer $N$ such that $p_{n-c_1}+p_{n-c_2}+\cdots +p_{n-c_g}>p_{n+d_1}+\cdots +p_{n+d_h}$ for all $n\ge N$. We prove this result using only the prime number theorem. For any instance of this result, we sketch a way to find the least possible $N$. We give some numerical results and unanswered questions.