Problem 2

Let

a_1, a_2, a_3, \ldots

be an infinite increasing sequence of positive integers. Prove that for every

p \ge 1

there are infinitely many

a_m

which can be written in the form

a_m = xa_p + ya_q

with

x, y

positive integers and

q > p

Step 4 of 4: Conclude: infinitely many representable terms

Previous step Next step

\{a_{i_k}\}_{k\ge 2} \text{ infinite} \implies \text{infinitely many } a_m=xa_p+ya_q

Detailed analysis

Step 3 applies to every $k \ge 2$ , and the subsequence $(a_{i_k})_{k \ge 1}$ is infinite, so there are infinitely many indices $m = i_k$ ( $k \ge 2$ ) for which $a_m = xa_p + ya_q$ with positive integers $x, y$ and $q = i_1 > p$ . Since $p \ge 1$ was arbitrary, this holds for every $p$ , which is exactly the statement to be proved. $\blacksquare$