Conjectures on Sums of Consecutive Primes

Abstract

We study additive properties of consecutive prime numbers and the primality of the sums they generate. For a given prime number $p_n$, we consider the sums \[ S_k(p_n) = p_n + p_{n+1} + \cdots + p_{n+k-1}, \] where $k \ge 3$ is an odd integer. We first formulate an existence conjecture asserting that, for every prime number $p_n$, there exists at least one odd length $k \ge 3$ such that $S_k(p_n)$ is itself a prime number. An exhaustive computational verification covering the first one million prime numbers revealed no counterexamples. We then propose a strengthened conjecture according to which, for every prime number $p_n$, there exist infinitely many odd lengths $k$ such that $S_k(p_n)$ is prime. This strong version is supported by a probabilistic heuristic showing that the series of the corresponding primality probabilities diverges, suggesting that the phenomenon is not exceptional but recurrent. We also analyze the possible modular obstructions, showing that they are local in nature and cannot persist when the length $k$ varies among odd integers. A Diophantine interpretation of the problem is proposed, together with a conceptual comparison with the generalized Goldbach conjecture. Finally, we discuss the role of the Generalized Riemann Hypothesis (GRH) in controlling the distribution of the sums under consideration. These structural, modular, Diophantine, and probabilistic (heuristic) arguments support both conjectures and formalize heuristic theorems of Cram\'er, GRH, and Hardy--Littlewood type explaining the expected absence of counterexamples.