Polignac's Conjecture Explorer

What is Polignac's Conjecture?

Polignac's Conjecture (1849) states that for every positive even integer $k$, there are infinitely many prime gaps of size exactly $k$. In other words:

$$\text{For each even } k > 0, \text{ there exist infinitely many primes } p \text{ such that } p + k \text{ is also prime}$$

Historical Context: The most famous case is $k = 2$, known as the Twin Prime Conjecture. Examples include (3,5), (5,7), (11,13), (17,19), (29,31), (41,43), etc. Despite extensive computational verification and strong probabilistic evidence, no proof exists for any specific value of $k$.

                    Current Status: While unproven, Zhang's 2013 breakthrough showed that there are infinitely many prime gaps bounded by 70 million (later improved to 246). This doesn't prove Polignac's conjecture but demonstrates that bounded prime gaps occur infinitely often.
                

Interactive Gap Explorer

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Probabilistic Evidence

The Hardy-Littlewood k-tuple conjecture provides a probabilistic framework predicting the density of prime gaps:

$$\pi_k(x) \sim C_k \frac{x}{(\ln x)^2} \text{ as } x \to \infty$$

where $\pi_k(x)$ counts prime pairs $(p, p+k)$ with $p \leq x$, and $C_k$ is a constant depending on $k$.

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Computational Evidence & Counterexample Search

While we expect Polignac's conjecture to be true, let's search systematically for evidence and potential counterexamples:

Challenge: Can you find an even number $k$ for which NO prime gaps of size $k$ exist within our search range? (Spoiler: you probably won't for reasonable values, which supports the conjecture!)

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Why Mathematicians Believe Polignac's Conjecture

✅ Supporting Evidence

Computational verification for gaps up to hundreds in large ranges
Probabilistic models predict infinite occurrences
Hardy-Littlewood conjecture provides theoretical framework
Random model strongly suggests truth
No counterexamples found despite extensive searching

❓ Challenges

No proof technique currently handles the general case
Prime distribution has subtle correlations
Computational limits can't verify infinity
Analytic methods fall short of complete proof
Technical barriers similar to other major conjectures

🎯 The Probabilistic Argument

If primes were distributed "randomly" with density $\frac{1}{\ln n}$ near $n$, then the probability that both $n$ and $n+k$ are prime is approximately $\frac{1}{(\ln n)^2}$. Summing this over all $n$ gives a divergent series, suggesting infinitely many such pairs exist.

This heuristic argument isn't a proof because primes aren't truly random, but it provides compelling evidence for why the conjecture should be true.

🔢 Polignac's Conjecture Explorer