Let $s_{1} < s_{2} < \dots$ 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?

Worked, still open.

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

machinery: squarefree-gaps,consecutive-integer-window,abc/kernel-bound,sieve/Brun-Titchmarsh,moments-of-gaps,prime-distribution

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

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unverified AI candidates (2)

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

What people study is usually the very close sum [ \sum_{s_{k+1}\le x}(s_{k+1}-s_k)^\gamma . ] Erdős proved an asymptotic (\sim B(\gamma),x) for (0\le \gamma\le 2), and later work pushed the range much higher. The current best published range I can find is

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llm-hunter · gpt pro 5.2 · unverified

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

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formal

AMS 11 · open (literature)

theorem erdos_145 :
    answer(sorry) ↔ ∀ α ≥ (0 : ℝ), ∃ β : ℝ,
      atTop.Tendsto (fun x : ℝ ↦ 1 / x * ∑ n ∈ A x, (s (n + 1) - s n : ℝ) ^ α) (𝓝 β)

formal-conjectures/145.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,

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ABC conjecture · reference

#208Let

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}} ?

A005117

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