Questions tagged [triangular-numbers]

Numbers of the form $\sum_{i=1}^ni=1+2+3+4+...+n$.

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Sum of first n terms is prime.

$a_n$ is a non-decreasing sequence of positive integers. If an positive integer $k$ appears in $a_n$ exactly $k$ times and $S_n$ is the sum of the first n terms, find all $N$ such that $S_N$ is prime. ...
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If $T_n$ is the $n$-th triangular number, there are an infinite number of $a, b, c, d$ such that $T_n+T_{an+b} =(cn+d)^2 $ for all $n$.

If $T_n$ is the $n$-th triangular number, show that there are an infinite number of positive integers $a, b, c, d$ such that $T_n+T_{an+b} =(cn+d)^2 $ for all $n$. This is inspired by an article in ...