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 2 of 5: Reduce every term

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\gcd(K^2+i,K)=i,\quad q_i=\frac{a_i}{b_i},\quad a_i=\frac{K^2+i}{i},\ b_i=\frac Ki

Detailed analysis

Because i divides K and K^2, gcd(K^2+i,K)=gcd(i,K)=i. Thus the reduced numerator and denominator are a_i=(K^2+i)/i and b_i=K/i, which are relatively prime positive integers.