Let $s_{1} < s_{2} < \dots$ be the sequence of squarefree numbers. Is it true that, for any $ϵ > 0$ and large $n$ , $s_{n + 1} - s_{n} ≪_{ϵ} s_{n}^{ϵ} ?$ Is it true that $s_{n + 1} - s_{n} \leq (1 + o (1)) \frac{π ^{2}}{6} \frac{lo g s _{n}}{lo g lo g s _{n}} ?$

Worked, still open.

number theory · open · formalized (Lean) · 0 attempts

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Both inequalities are **open unconditionally**. What is known is (roughly) that we can prove only **power–type** upper bounds on the maximal gap, while Erdős already proved in 1951 that gaps of the size in your second display occur infinitely often (so if an upper bound of that form is true, the constant would be best …

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

candidate solution ↗

formal

AMS 11 · open (literature)

theorem erdos_208.parts.i : answer(sorry) ↔
    ∀ ε > (0 : ℝ), (fun n => (s (n + 1) - s n : ℝ)) =O[atTop] (fun n => (s n : ℝ)^ε)

formal-conjectures/208.lean ↗

oeis

A005117 — Squarefree numbers: numbers that are not divisible by a square greater than 1.1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,30,31,33,34,35,37,38,39,41,42,43,46,47,51,53,55,57,58,59,61,62,65,66,67,A076259 — Gaps between squarefree numbers: a(n) = A005117(n+1) - A005117(n).1,1,2,1,1,3,1,2,1,1,2,2,2,1,1,3,3,1,1,2,1,1,2,1,1,2,1,1,3,1,4,2,2,2,1,1,2,1,3,1,1,2,1,1,2,1,3,1,1,3,1,2,1,1,2,2,2,1,1,2,

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ABC conjecture · reference

#145Let

s_{1} < s_{2} < \dots

be the sequence of squarefree numbers. Is it true that, for any

α \geq 0

x \to \infty lim \frac{1}{x} s_{n} \leq x \sum (s_{n + 1} - s_{n})^{α}

exists?A005117

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