erdős #770

← #769 · #771 → (packet.json; erdosproblems.com)

Let $h (n)$ be minimal such that $2^{n} - 1, 3^{n} - 1, \dots, h (n)^{n} - 1$ are mutually coprime. Does, for every prime $p$ , the density $δ_{p}$ of integers with $h (n) = p$ exist? Does $lim inf h (n) = \infty$ ? Is it true that if $p$ is the greatest prime such that $p - 1 ∣ n$ and $p > n^{ϵ}$ then $h (n) = p$ ?

Worked, still open.

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

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unverified AI candidates (2)

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

I’ll take “mutually coprime” in the (standard in this Erdős-problem context) sense that the **whole set has gcd 1**, i.e. [ h(n):=\min\\{m\ge 2:\gcd\bigl(2^n-1,3^n-1,\dots,m^n-1\bigr)=1\\}, ] which is exactly the formulation of Erdős Problem #770. ([Erdős Problems][1]) (If one interpreted “mutually coprime” as *pairwis…

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

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

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formal

AMS 11 · textbook (literature)

theorem Nat.Prime.h_eq_add_one {n : ℕ} (hn : 2 < n) : h n = n + 1 ↔ (n + 1).Prime

formal-conjectures/770.lean ↗

oeis

A263647 — Numbers k such that 2^k-1 and 3^k-1 are coprime.1,2,3,5,7,9,13,14,15,17,19,21,25,26,27,29,31,34,37,38,39,41,45,47,49,51,53,57,59,61,62,63,65,67,71,73,74,79,81,85,87,89,

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#820Let

H (n)

be the smallest integer

l

such that there exist

k < l

with

(k^{n} - 1, l^{n} - 1) = 1

.Is it true that

H (n) = 3

infinitely often? (That is,

(2^{n} - 1, 3^{n} - 1) = 1

infinitely often?)Estimate

H (n)

. Is it true that there exists some constant

c > 0

such that, for all

ϵ > 0

H (n) > exp (n^{(c - ϵ) / l o g l o g n})

for infinitely many

n

and

H (n) < exp (n^{(c + ϵ) / l o g l o g n})

for all large enough

n

k

such that

(k^{n} - 1, 2^{n} - 1) = 1

?A263647

status