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Week 19 · ClassicalOpen

Primary source

Andrica's Conjecture — How Tight Are Prime Gaps?

Simple rule · open problem· Posed 1986

Problem

Does $\sqrt{p_{n+1}} - \sqrt{p_n} < 1$ hold for every $n \ge 1$ , where $p_n$ is the $n$ th prime?

Why it matters

Equivalent form: $p_{n+1} - p_n < 2\sqrt{p_n} + 1$ . By PNT, average gaps are $\sim \ln p_n$ , so $\sqrt{p_n}$ feels comfortable — but the worst-case bound is what's in question.

Progress so far

Verified for $p \le 1.32 \times 10^{16}$ (Ghory 2010). The maximum value occurs at $n=4$ : $\sqrt{11}-\sqrt{7} \approx 0.6709$ . Even RH does not directly imply it — Cramér or Oppermann conjectures would.

💡 Explore together, one line at a time(0 contributions)

Contributions are not ranked by popularity. Curator feedback names what is clear or reproducible, and peer signals mean someone understood or actually reproduced it.

What did you notice?

You do not need a complete proof. A small observation can open the next path.

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