Triangular Number Pairs with Triangular Sum and Difference/Mistake

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Source Work

1997: David Wells: Curious and Interesting Numbers:

The Dictionary
$15$


1997: David Wells: Curious and Interesting Numbers (2nd ed.):

The Dictionary
$15$


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 $2,185,095$.


Correction

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 Leonard Eugene Dickson in his $1920$ work History of the Theory of Numbers, Volume II:

J. Ozanam 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 $2090$.

Jacques Ozanam's work dates from the $17$th century, and has been considerably superseded by further research. In particular, many other such pairs have since been discovered, rendering Wells's thesis considerably out of date.

Indeed, Dickson himself goes on to say, in a footnote:

Others are $171$ and $105$, $3741$ and $2145$.


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


Sources