Conjectures on Sums of Consecutive Primes

TL;DR

The paper investigates sums of $k$ consecutive primes, $S_k(p_n)=\sum_{i=0}^{k-1} p_{n+i}$, with odd $k\ge3$, and conjectures that for every prime $p_n$ there exists at least one such $k$ making $S_k(p_n)$ prime; computational checks up to the $10^6$-th prime reveal no counterexample. It develops conditional support via the Cramér model and GRH, showing that the probability of never obtaining a prime sum across odd lengths tends to zero and that modular obstructions are unlikely to persist as $k$ varies. A Diophantine viewpoint and a local-global heuristic are used to reason about why sums of consecutive primes should frequently yield primes, drawing connections to the generalized Goldbach problem. The paper also strengthens the conjecture to assert that infinitely many odd lengths $k$ exist for each $p_n$ with $S_k(p_n)$ prime, providing concrete algorithmic examples and arguing that, under standard probabilistic assumptions, counterexamples are exceedingly improbable. Overall, the work integrates numerical evidence, modular and Diophantine reasoning, and probabilistic heuristics to justify the plausibility of both conjectures and to relate them to broader themes in additive prime problems.

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 Cramér, GRH, and Hardy--Littlewood type explaining the expected absence of counterexamples.

Theorems & Definitions (5)

• Conjecture 1: Consecutive Prime Sums
• Theorem 1: Cramér-type Heuristic
• Theorem 2: Conditional under GRH
• Theorem 3: Heuristic
• Conjecture 2: Infinite admissible lengths