Erdős Problem 678

Reference: erdosproblems.com/678

def lcmInterval (n k : ℕ) : ℕ

Write $M(n, k)$ be the least common multiple of ${n+1, \dotsc, n+k}$, denoted here as lcmInterval n k.

Equations

• lcmInterval n k = (Finset.Ioc n (n + k)).lcm id

theorem lcmInterval_lt_example1 : lcmInterval 104 8 < lcmInterval 96 7

The referee of [Er79] found the example $M(96, 7) > M(104, 8)$, showing that there are cases where $M(n, k) > M(m, k + 1)$ with $m \geq n + k$. [Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.

theorem lcmInterval_lt_example2 : lcmInterval 139 8 < lcmInterval 132 7

The referee of [Er79] found the example $M(132, 7) > M(139, 8)$, showing that there are cases where $M(n, k) > M(m, k + 1)$ with $m \geq n + k$. [Er79] Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.

theorem lcmInterval_lt_example3 : lcmInterval 62 8 < lcmInterval 52 7

Cambie [Ca24] found the example $M(52, 7) > M(62, 8)$. [Ca24] S. Cambie, Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138 (2024).

theorem lcmInterval_lt_example4 : lcmInterval 47 9 < lcmInterval 36 8

Cambie [Ca24] found the example $M(36, 8) > M(48, 9)$. [Ca24] S. Cambie, Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138 (2024).

theorem erdos_678 : (∀ (k : ℕ), 3 ≤ k → {(m, n) : ℕ × ℕ | n + k ≤ m ∧ lcmInterval m (k + 1) < lcmInterval n k}.Infinite) ↔ True

Write $M(n, k)$ be the least common multiple of ${n+1, \dotsc, n+k}$. Let $k \geq 3$. Are there infinitely many $m, n$ with $m \geq n + k$ such that $$ M(n, k) > M(m, k + 1) $$? The answer is yes, as proved in a strong form by Cambie [Ca24]. [Ca24] S. Cambie, Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138 (2024).