Triangular Number Pairs with Triangular Sum and Difference/Mistake

Source Work

 * The Dictionary
 * $15$
 * $15$

Mistake

 * $15$ and $21$ are the smallest pair of triangular numbers whose sum and difference ($6$ and $36$) are also triangular. The next such pairs are $780$ and $990$, and $1,747,515$ and $2,185,095$.

There are a good number of other such pairs in addition to these. It can be noted that $780$ and $990$ are the $4$th such pair, and that $1,747,515$ and $2,185,095$ are in fact the $26$th (in ascending order of the larger element of the pairs).

The pairs given are those reported by in his $1920$ work :


 *  found pairs$^*$ of triangular numbers $15$ and $21$, $780$ and $990$, $1747515$ and $2185095$, whose sum and difference are triangular. Their sides are $5$ and $6,$ $39$ and $44$, $1869$ and $2090$.

's work dates from the $17$th century, and has been considerably superseded by further research. In particular, many other such pairs have since been discovered, rendering 's thesis considerably out of date.

Indeed, himself goes on to say, in a footnote:
 * Others are $171$ and $105$, $3741$ and $2145$.

repeats this mistake in $2$ further places: in section $780$ and again in section $1,747,515$.