erdős #687

← #686 · #688 → (packet.json; erdosproblems.com)

Let $Y (x)$ be the maximal $y$ such that there exists a choice of congruence classes $a_{p}$ for all primes $p \leq x$ such that every integer in $[1, y]$ is congruent to at least one of the $a_{p} (mod p)$ . Give good estimates for $Y (x)$ . In particular, can one prove that $Y (x) = o (x^{2})$ or even $Y (x) ≪ x^{1 + o (1)}$ ?

Worked, still open.

number theory · open · prize $1000 · 0 attempts

use this record

vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

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

Let [ P(x):=\prod_{p\le x}p ] [[nomath]](the primorial up to $x$)[[/nomath]]. Your definition is exactly the “Erdős–Rankin/Jacobsthal” quantity that appears in large prime gap constructions.

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

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

candidate solution ↗

oeis

A048670 — Jacobsthal function A048669 applied to the product of the first n primes (A002110).2,4,6,10,14,22,26,34,40,46,58,66,74,90,100,106,118,132,152,174,190,200,216,234,258,264,282,300,312,330,354,378,388,414,4 A058989 — Largest number of consecutive integers such that each is divisible by a prime <= the n-th prime.1,3,5,9,13,21,25,33,39,45,57,65,73,89,99,105,117,131,151,173,189,199,215,233,257,263,281,299,311,329,353,377,387,413,431

links

Create a formalisation here · link

#854Let

n_{k}

denote the

k

th primorial, i.e. the product of the first

k

primes.If

1 = a_{1} < a_{2} < \dots a_{ϕ (n_{k})} = n_{k} - 1

is the sequence of integers coprime to

n_{k}

, then estimate the smallest even integer not of the form

a_{i + 1} - a_{i}

. Are there

≫ i max (a_{i + 1} - a_{i})

many even integers of the form

a_{j + 1} - a_{j}

?A048670 #860Let

h (n)

be such that, for any

m \geq 1

, in the interval

(m, m + h (n))

there exist distinct integers

a_{i}

for

1 \leq i \leq π (n)

such that

p_{i} ∣ a_{i}

, where

p_{i}

denotes the

i

th prime. Estimate

h (n)

.A048670

status

open

evidence

oeis

links

status

Search Vela