✦ 5 frontiers

read-only vieweressays

erdős #374

← #373 · #375 → (packet.json; erdosproblems.com)

For any $m \in N$ , let $F (m)$ be the minimal $k \geq 2$ (if it exists) such that there are $a_{1} < \dots < a_{k} = m$ with $a_{1}! \dots a_{k}!$ a square. Let $D_{k} = {m : F (m) = k}$ . What is the order of growth of $∣ D_{k} \cap {1, \dots, n}∣$ for $3 \leq k \leq 6$ ? For example, is it true that $∣ D_{6} \cap {1, \dots, n}∣ ≫ n$ ?

Worked, still open.

number theory · open · 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

Write [ D_k(n):=\lvert D_k\cap{1,\dots,n}\rvert. ]

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

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

candidate solution ↗

oeis

A387184 — Numbers d such that a!*b!*c!*d! is a perfect square for some 1<=a<b<c<d.6,8,9,10,12,14,15,16,18,20,21,22,24,25,26,27,28,30,32,33,34,35,36,38,39,40,42,44,45,48,49,50,51,52,54,55,56,57,60,62,63,A388851 — Numbers c such that a! * b! * c! is a perfect square for some 1 <= a < b < c.4,6,8,9,10,16,18,20,24,25,28,30,32,35,36,45,49,50,54,63,64,70,72,77,80,81,96,98,100,112,120,121,125,126,128,140,144,150,A389117 — Decimal expansion of the sum of the distinct entries of 1/A055204(n)^(1/2).3,7,0,9,7,5,1,2,3,3,8,9,9,2,2,2,3,6,8,8,8,7,9,6,1,4,6,6,8,4,2,0,7,8,8,8,2,1,7,4,4,3,6,4,5,0,6,4,1,6,9,7,6,6,8,0,9,7,8,9,A389148 — Composite numbers e>6 for which there is no 1<=a<b<c<d<e for which a!b!c!d!e! is a perfect square.527,611,713,731,779,893,923,1003,1037,1271,1273,1343,1349,1357,1411,1469,1591,1643,1679,1781,1919,1927,1943,1957,2033,20

links

Create a formalisation here · link

status

open

finding.noted · reviewer:will-blair · 1 day

renders the record as of vev_d199cb2e · 1,338 events · hub