Problem 4

Prove that for any positive integer

k

, there exists an arithmetic sequence

\frac{a_1}{b_1},\frac{a_2}{b_2},\ldots,\frac{a_k}{b_k}

of rational numbers, where

a_i,b_i

are relatively prime positive integers for each

i=1,2,\ldots,k

, such that the positive integers

a_1,b_1,a_2,b_2,\ldots,a_k,b_k

are all distinct.

Step 1 of 5: Choose the arithmetic sequence

K=k!,\quad q_i=\frac{K^2+i}{K}\quad(1\le i\le k)

Detailed analysis

For k=1 the assertion is immediate. For k≥2 put K=k! and consider the arithmetic sequence q_i=(K^2+i)/K for i=1,…,k; its common difference is 1/K.