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Conjectures on Sums of Consecutive Primes

This paper proposes and supports two conjectures asserting that for every prime number, there exists at least one and potentially infinitely many odd-length sums of consecutive primes starting from that number which are themselves prime, a claim backed by extensive computational verification, probabilistic heuristics, and modular analysis.

Original authors: Edwige Tolla

Published 2026-01-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Edwige Tolla

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a long line of people, but instead of names, they are numbered with prime numbers (numbers like 2, 3, 5, 7, 11, 13... that can only be divided by 1 and themselves). These people are standing in perfect order, one after another.

The paper by Edwige Tolla asks a very specific question about these people: If you start with any single person in this line and ask them to grab the hands of the next few people to form a group, can you always find a group size that makes the "total weight" of the group a prime number?

Here is the breakdown of the paper's ideas using simple analogies:

1. The Game Rules

The Line: You have an endless line of prime numbers ( $p_n, p_{n+1}, p_{n+2}, \dots$ ).
The Group: You pick a starting person ( $p_n$ ) and grab the next $k$ people.
The Constraint: The group size ( $k$ ) must be an odd number (3, 5, 7, etc.) and at least 3 people long.
The Goal: Add up the numbers of everyone in the group. Is the total sum a prime number?

The Paper's Main Claim (Conjecture 1):
No matter which person you start with in the line, you can always find a group size (an odd number) that makes the total sum a prime number.

2. The "Search Party" (Computational Results)

The author didn't just guess; they built a computer program to play this game.

They tested the first one million starting people in the line.
For every single one of them, the computer found a group size that worked.
The Result: They found zero failures. Not a single person in that first million couldn't find a matching group.

Analogy: Imagine searching for a specific key in a giant library. You check one million different books, and in every single one, you find the key hidden on a specific page. You haven't found a book without a key yet.

3. Why Does This Happen? (The Probabilistic View)

The paper uses math models (like Cramér's model) to explain why this probably works, even though they haven't proven it with absolute certainty yet.

The Lottery Analogy: Think of checking different group sizes (3, 5, 7, 9...) as buying lottery tickets.
The Odds: The odds of a random large number being prime are roughly 1 in the size of the number's "logarithm" (a way of measuring how big it is).
The Infinite Tickets: Even if the odds of winning on the first try (group size 3) are low, you get to keep buying tickets by trying group sizes 5, 7, 9, and so on.
The Conclusion: Because you have an infinite supply of tickets (group sizes) and the odds don't drop to zero, the math suggests that eventually, you must win. The paper argues that the chance of never winning is effectively zero.

4. The "Traffic Jam" Argument (Modular Obstructions)

Sometimes, math problems get stuck because of "traffic jams" (modular obstructions). For example, if you add three odd numbers, the result is always odd. If you add numbers in a way that always results in a multiple of 3, you can never get a prime (except for 3 itself).

The Paper's Finding: The author checked if there is a "traffic jam" that forces the sum to be divisible by a small number (like 3 or 5) no matter how big the group gets.
The Result: There is no permanent traffic jam. As you change the group size, the "traffic" clears up. The sums eventually land in "open lanes" where they can be prime.

5. The Stronger Claim (Conjecture 2)

The paper doesn't just stop at finding one working group size. It suggests something even more amazing:

The Claim: For any starting person, there aren't just one or two group sizes that work; there are infinitely many.
The Evidence: When the author looked at specific examples, they found dozens of different group sizes that worked for the same starting number.
The Logic: Since the "lottery tickets" keep coming and the odds remain favorable, you shouldn't just win once; you should win over and over again forever.

6. Connection to Famous Problems

The author compares this to the Goldbach Conjecture (a famous unsolved problem stating that every even number greater than 2 is the sum of two primes).

Goldbach: You can pick any two primes you want to make a sum.
This Paper: You are forced to pick primes that are neighbors in the line.
The Twist: Even though this rule is much stricter (harder), the fact that you can change the length of the group (the number of neighbors) gives you enough flexibility to make it work, just like Goldbach.

Summary

The paper proposes a new rule about prime numbers: If you start anywhere in the line of primes and add up an odd number of neighbors, you will eventually find a sum that is also a prime.

The author has checked the first million cases and found no exceptions. Using probability math and logic about how numbers are distributed, they argue that this rule is likely true for all prime numbers, and that there are likely infinite ways to make it work for any starting point. While it's not a formal proof yet, the evidence is very strong.

Technical Summary: Conjectures on Sums of Consecutive Primes

Problem Statement
The paper investigates the additive properties of consecutive prime numbers, specifically focusing on the primality of their partial sums. Let $p_n$ denote the $n$ -th prime number. The study defines the sum of $k$ consecutive primes starting from $p_n$ as:
$S_k(p_n) = p_n + p_{n+1} + \dots + p_{n+k-1}$
The central problem is to determine, for a given starting prime $p_n$ , whether there exists an odd integer length $k \ge 3$ such that $S_k(p_n)$ is itself a prime number. The paper distinguishes between two levels of inquiry: the existence of at least one such $k$ (Existence Conjecture) and the existence of infinitely many such $k$ (Strengthened Conjecture).

Methodology
The authors employ a multi-faceted approach combining computational verification, probabilistic modeling, modular analysis, and Diophantine interpretation:

Computational Verification: An algorithm was implemented to test the first $10^6$ prime numbers ( $p_{10^6} = 15,485,863$ ). For each $p_n$ , the algorithm iteratively checks odd lengths $k = 3, 5, 7, \dots$ until a prime sum $S_k(p_n)$ is found.
Probabilistic Heuristics: The study utilizes Cramér's probabilistic model and the Hardy–Littlewood conjectures. It treats the primality of $S_k(p_n)$ as a stochastic event with probability approximately $1/\ln(S_k(p_n))$ . The authors analyze the divergence of the series of these probabilities to argue that the event of finding a prime sum is recurrent rather than exceptional.
Modular and Diophantine Analysis: The paper examines potential modular obstructions (where $S_k(p_n)$ might be forced to be divisible by small primes). It argues that such obstructions are local and depend on specific values of $k$ ; as $k$ varies over odd integers, these obstructions do not persist globally. The problem is framed as a constrained Diophantine equation where the solution space is sparse but not obstructed by global modular constraints.
Conditional Frameworks: The analysis is contextualized within the Generalized Riemann Hypothesis (GRH). Under GRH, the distribution of primes in arithmetic progressions is sufficiently uniform to suggest that no systematic arithmetic obstruction prevents $S_k(p_n)$ from being prime for some odd $k$ .

Key Contributions and Results

Conjecture 1 (Existence): The paper formulates the conjecture that for every prime $p_n$ , there exists at least one odd integer $k \ge 3$ such that $S_k(p_n)$ is prime.
- Result: Exhaustive computational verification up to $n = 10^6$ yielded no counterexamples. The algorithm successfully identified a minimal odd $k$ for every tested prime.
- Observation: The minimal length $k_{min}(p_n)$ required to find a prime sum appears to grow logarithmically with $p_n$ (specifically $k_{min} \propto 2 \ln p_n$ ), consistent with geometric distribution models.
Conjecture 2 (Infinite Admissible Lengths): A strengthened version posits that for every $p_n$ , there exist infinitely many odd integers $k \ge 3$ such that $S_k(p_n)$ is prime.
- Result: Numerical examples for specific primes (e.g., $p_2=3, p_3=5, p_4=7$ ) show multiple valid $k$ values within the range $k < 1000$ . For instance, for $p_{651511}$ , 32 distinct odd lengths up to 999 yielded prime sums.
- Heuristic Support: The authors argue that since the series $\sum_{k \text{ odd}} \frac{1}{\ln(k p_n)}$ diverges, the Borel–Cantelli lemma (applied heuristically) suggests that the event " $S_k(p_n)$ is prime" occurs infinitely often.
Theoretical Justification:
- Theorem 1 (Cramér-type): Under Cramér's model, the probability that a prime $p_n$ exists with no odd-length prime sum is zero.
- Theorem 2 (Conditional on GRH): Assuming the Generalized Riemann Hypothesis, for every sufficiently large $p_n$ , such a $k$ exists.
- Theorem 3 (Heuristic): Under standard Hardy–Littlewood heuristics, the probability that all odd sums starting from a given $p_n$ are composite is zero.

Significance and Claims
The paper claims to provide a "strong body of evidence" supporting both conjectures through the convergence of numerical data, modular arguments, and probabilistic heuristics.

Conceptual Contribution: The work situates the problem within the family of additive prime problems, drawing a conceptual parallel to the Generalized Goldbach Conjecture. However, it highlights a key distinction: while Goldbach allows free choice of primes, this problem imposes strict consecutivity constraints. The authors argue that the variability of the sum length $k$ compensates for this rigidity, effectively eliminating local modular obstructions.
Formalization: The paper formalizes heuristic arguments (Cramér, Hardy–Littlewood, GRH) to explain the expected absence of counterexamples, framing the problem as a Diophantine equation where the lack of persistent local obstructions implies the existence of global solutions.
Modesty of Claims: The authors explicitly state that their results are heuristic and conditional. They do not claim a rigorous proof of the conjectures but rather a formalization of why counterexamples are expected to be non-existent or extremely rare. The computational results are presented as verification up to $10^6$ , not as a proof for all integers.

In summary, the paper proposes that the set of sums of consecutive primes of odd length is rich enough to contain a prime number for every starting point, and likely infinitely many such primes, a phenomenon supported by the divergence of primality probabilities and the absence of global modular obstructions.