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.