MathDB

f(n) < 3\sqrt{n} and f(n) > k if n > k! + k, L(n, 1) < L(n, 2) < ... < L(n, k)

Source: Indian Postal Coaching 2009 set 3 p5

May 26, 2020

number theoryleast common multipleLCMinequalities

Problem Statement

For positive integers

n, k

with

1 \le k \le n

, define

L(n, k) = Lcm \,(n, n - 1, n -2, ..., n - k + 1)

Let

f(n)

be the largest value of

k

such that

L(n, 1) < L(n, 2) < ... < L(n, k)

. Prove that

f(n) < 3\sqrt{n}

and

f(n) > k

n > k! + k