User:Wguynes/Proofs

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Derivation of triangular number test function

Q: What triangular numbers are also tetrahedral numbers?

Let Tn = nth triangular number.

Let Tm = mth tetrahedral number.

Tn = Tm where

\left\{ n = \frac{-3+\sqrt{12m(m+1)(m+2)+9}}{6} \quad | \quad n \in \Z^+ , m \in \Z^+ \right\}
\left\{ n = \sqrt{\tfrac{1}{3}m^3+m^2+\tfrac{2}{3}m+\tfrac{1}{4}} - \tfrac{1}{2} \quad | \quad n \in \Z^+ , m \in \Z^+ \right\}