Is it true that, for every integer $t \geq 1$ , there is some integer $a$ such that $(k n) = a$ (with $1 \leq k \leq n /2$ ) has exactly $t$ solutions?

Worked, still open.

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

machinery: binomial-coefficient-multiplicity,Singmaster-conjecture,elliptic-integral-points,Diophantine-equation-binomials,prime-distribution,Kummer-carries,consecutive-integer-window

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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

Let [ N(a)=|\\{(n,k)\in\mathbb Z_{\ge 1}^2:\ 1\le k\le n/2,\ \binom{n}{k}=a\\}|. ] Your question asks whether **every** (t\ge 1) occurs as $N(a)$ for some integer $a$.

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_849 : answer(sorry) ↔
    ∀ t ≥ 1, ∃ a : ℕ,
      {n : ℕ | ∃ k ≥ 1, 2 * k ≤ n ∧ choose n k = a}.ncard = t

formal-conjectures/849.lean ↗

oeis

A003015 — Numbers that occur 5 or more times in Pascal's triangle.1,120,210,1540,3003,7140,11628,24310,61218182743304701891431482520 A003016 — Number of occurrences of n as an entry in rows <= n of Pascal's triangle (A007318).0,3,1,2,2,2,3,2,2,2,4,2,2,2,2,4,2,2,2,2,3,4,2,2,2,2,2,2,4,2,2,2,2,2,2,4,4,2,2,2,2,2,2,2,2,4,2,2,2,2,2,2,2,2,2,4,4,2,2,2,A059233 — Number of rows in which n appears in Pascal's triangle A007318.1,1,1,1,2,1,1,1,2,1,1,1,1,2,1,1,1,1,2,2,1,1,1,1,1,1,2,1,1,1,1,1,1,2,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,2,2,1,1,1,1,1,A090162 — Values of binomial(Fibonacci(2k)*Fibonacci(2k+1),Fibonacci(2k-1)*Fibonacci(2k)-1).1,3003,61218182743304701891431482520 A098565 — Numbers that appear as binomial coefficients exactly 6 times.120,210,1540,7140,11628,24310,61218182743304701891431482520 A180058 — Smallest number occurring in exactly n rows of Pascal's triangle.2,6,120,3003 A182237 — Numbers occurring exactly in 2 rows of Pascal's triangle.6,10,15,20,21,28,35,36,45,55,56,66,70,78,84,91,105,126,136,153,165,171,190,220,231,252,253,276,286,300,325,330,351,364,3

links

Singmaster's conjecture · reference

status