## Triangular numbers : T_{A} * T_{B} = T_{C^2 + C}

$T_{A} \cdot T_{B} \; = \; T_{C^2 + C}$

where   $A, \; B \; > \; 1$

$T_6 \cdot T_4 \; = \; T_{5^2 - 5} \; = \; T_{4^2 + 4}$
$21 \times 10 \; = \; 210 \; = \; T_{20}$

$T_{13} \cdot T_9 \; = \; T_{10^2 - 10} \; = \; T_{9^2 + 9}$
$91 \times 45 \; = \; 4095 \; = \; T_{90}$

There are infinite many solutions of the form,
$T_{A} \cdot T_{B} \; = \; T_{B^2 + B} \; = \; T_{(B+1)^2 - (B+1)}$

$T_{78} \cdot T_{55} = T_{55^2 + 55} = T_{56^2 - 56}$
$T_{457} \cdot T_{323} = T_{323^2 + 323} = T_{324^2 - 324}$
$T_{2666} \cdot T_{1885} = T_{1885^2 + 1885} = T_{1886^2 - 1886}$
$T_{15541} \cdot T_{10989} = T_{10989^2 + 10989} = T_{10990^2 - 10990}$
$T_{90582} \cdot T_{64051} = T_{64051^2 + 64051} = T_{64052^2 - 64052}$
$T_{527953} \cdot T_{373319} = T_{373319^2 + 373319} = T_{373320^2 - 373320}$
$T_{3077138} \cdot T_{2175865} = T_{2175865^2 + 2175865} = T_{2175866^2 - 2175866}$
$T_{17934877} \cdot T_{12681873} = T_{12681873^2 + 12681873} = T_{12681874^2 - 12681874}$
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math grad - Interest: Number theory
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### One Response to Triangular numbers : T_{A} * T_{B} = T_{C^2 + C}

1. paul says:

Here are a few more

T37 . T26 = T(27^2 – 27)
703 x 351 = 246753 = T702

T51 . T7 = T(17^2 – 17)
1326 x 28 = 37128 = T272

T76 . T2 = T(12^2 – 12)
2926 x 3 = 8778 = T132

T78 . T55 = T(56^2 – 56)
3081 x 1540 = 4744740 = T3080

Paul.