Get ready Part 9

Algebra Level 2

Is there exist an infinite number of ordered pairs $(a,b)$ of integers such that for every positive integer $t$ , the number $at+b$ is a triangular number if and only if $t$ is a triangular number. (The triangular numbers are the $t_n=\frac{n(n+1)}{2}$ with $n$ in $\{0,1,2,...\}$ .)

Yes No

1 solution

Otto Bretscher
Oct 29, 2018

First note that $n$ is a triangular number iff $8n+1$ is a perfect square.

Now, if $b$ is any triangular number and $a=8b+1$ , a perfect square, then $t$ will be a triangular number iff $r=at+b$ is, since $8r+1=a(8t+1)$ .

Get ready Part 9

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