Burton, in "Elementary Number Theory", states that the following problems are due to Euler 1775:

If $n$ is a triangular number, then so are $9n+1$, $25n+3$ and $49n + 6$.

R. F. Jordan in the J. of Recreational Mathematics (1991, vol.23, p.78) proves the following generalization:

Let $t_k$ be the $k$-th triangular number. For all $k$ and all triangular numbers $n$, $(2k+1)^2n + t_k$ is also triangular.

and wonders whether Euler actually proved this generalization. Jordan wasn't the first to prove this, see for example the following proposed solution to the 49th Putnam 1988 question B6.

I looked through the Euler Archive, but didn't locate the suitable 1775 manuscript (either appearing in 1775, written in 1775, or presented in 1775).

What did Euler prove and where? What does Burton refer to?

EDIT: The only part of the question that remains unanswered is: As Euler didn't seem to prove this generalization, who was the first to notice it?

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    $\begingroup$ Jordan was certainly not the first to notice it if the result was not due to Euler. Problem B6 on the 1988 Putnam exam required the students to make this same observation. $\endgroup$ Dec 4, 2014 at 21:45
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    $\begingroup$ so $(2k+1)^2T_j + T_k=T_{2jk+j+k}$ $\endgroup$ Dec 7, 2014 at 12:51

1 Answer 1


Dickson, History of the Theory of Numbers, Volume II, page 12, writes,

L. Euler (pp. 264-5, about 1775) noted that $9\Delta_a+1=\Delta_{3a+1}$, $25\Delta_a+3=\Delta_{5a+2}$, $49\Delta_a+6=\Delta_{7a+3}$, $81\Delta_a+10=\Delta_{9a+4}$.

The reference Dickson gives is to Opera Postuma, 1, 1862.

  • $\begingroup$ Thanks! This is precisely the reference I was looking for. It answers the two main questions: what did Euler prove and where. I am still very interested in the answer to the third though. $\endgroup$
    – Ohad
    Dec 5, 2014 at 2:29
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    $\begingroup$ Are we really to believe that Euler, of all people, didn't notice the obvious generalization? "Opera Postuma" = posthumous works; this could have been a note to himself where he saw the pattern and saw no need to write down any more. $\endgroup$ Dec 5, 2014 at 4:43
  • $\begingroup$ I thought as much, but I'm not familiar enough with Euler's character to assume he will do so. $\endgroup$
    – Ohad
    Dec 5, 2014 at 15:34

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