# Triangular Number Pairs with Triangular Sum and Difference

## Theorem

The sequence of pairs of triangular numbers whose sum and difference are also both triangular begins:

$\tuple {15, 21}, \tuple {105, 171}, \tuple {378, 703}, \tuple {780, 990}, \tuple {1485, 4186}, \tuple {2145, 3741}, \tuple {5460, 6786}, \tuple {7875, 8778}$

The sequence of the first elements is A185129 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The sequence of the second elements is A185128 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Examples

### $T_5$ and $T_6$

The triangular numbers $T_5$ and $T_6$ have sum and difference which are themselves both triangular:

 $\ds T_5$ $=$ $\ds \frac {5 \times 6} 2$ $= 15$ $\ds T_6$ $=$ $\ds \frac {6 \times 7} 2$ $= 21$

 $\ds T_6 - T_5$ $=$ $\ds 21 - 15$ $\ds$ $=$ $\ds 6$ $\ds$ $=$ $\ds \dfrac {3 \times 4} 2$ $\ds$ $=$ $\ds T_3$

 $\ds T_6 + T_5$ $=$ $\ds 21 + 15$ $\ds$ $=$ $\ds 36$ $\ds$ $=$ $\ds \dfrac {8 \times 9} 2$ $\ds$ $=$ $\ds T_8$

$\blacksquare$

### $T_{14}$ and $T_{18}$

The triangular numbers $T_{14}$ and $T_{18}$ have sum and difference which are themselves both triangular:

 $\ds T_{14}$ $=$ $\ds \frac {14 \times 15} 2$ $= 105$ $\ds T_{18}$ $=$ $\ds \frac {18 \times 19} 2$ $= 171$

 $\ds T_{18} - T_{14}$ $=$ $\ds 171 - 105$ $\ds$ $=$ $\ds 66$ $\ds$ $=$ $\ds \dfrac {11 \times 12} 2$ $\ds$ $=$ $\ds T_{11}$

 $\ds T_{18} + T_{14}$ $=$ $\ds 171 + 105$ $\ds$ $=$ $\ds 276$ $\ds$ $=$ $\ds \dfrac {23 \times 24} 2$ $\ds$ $=$ $\ds T_{23}$

$\blacksquare$

### $T_{27}$ and $T_{37}$

The triangular numbers $T_{27}$ and $T_{37}$ have sum and difference which are themselves both triangular:

 $\ds T_{27}$ $=$ $\ds \frac {27 \times 28} 2$ $= 378$ $\ds T_{37}$ $=$ $\ds \frac {37 \times 38} 2$ $= 703$

 $\ds T_{37} - T_{27}$ $=$ $\ds 703 - 378$ $\ds$ $=$ $\ds 325$ $\ds$ $=$ $\ds \frac {25 \times 26} 2$ $\ds$ $=$ $\ds T_{25}$

 $\ds T_{37} + T_{27}$ $=$ $\ds 703 + 378$ $\ds$ $=$ $\ds 1081$ $\ds$ $=$ $\ds \frac {46 \times 47} 2$ $\ds$ $=$ $\ds T_{46}$

$\blacksquare$

### $T_{39}$ and $T_{44}$

The triangular numbers $T_{39}$ and $T_{44}$ have sum and difference which are themselves both triangular:

 $\ds T_{39}$ $=$ $\ds \frac {39 \times 40} 2$ $= 780$ $\ds T_{44}$ $=$ $\ds \frac {44 \times 45} 2$ $= 990$

 $\ds T_{44} - T_{39}$ $=$ $\ds 990 - 780$ $\ds$ $=$ $\ds 210$ $\ds$ $=$ $\ds \dfrac {20 \times 21} 2$ $\ds$ $=$ $\ds T_{20}$

 $\ds T_{44} + T_{39}$ $=$ $\ds 990 + 780$ $\ds$ $=$ $\ds 1770$ $\ds$ $=$ $\ds \dfrac {59 \times 60} 2$ $\ds$ $=$ $\ds T_{59}$

$\blacksquare$

### $T_{1869}$ and $T_{2090}$

The triangular numbers $T_{1869}$ and $T_{2090}$ have sum and difference which are themselves both triangular:

 $\ds T_{1869}$ $=$ $\ds \frac {1869 \times 1870} 2$ $\ds = 1 \, 747 \, 515$ $\ds T_{2090}$ $=$ $\ds \frac {2090 \times 2091} 2$ $\ds = 2 \, 185 \, 095$

 $\ds T_{2090} - T_{1869}$ $=$ $\ds 2 \, 185 \, 095 - 1 \, 747 \, 515$ $\ds$ $=$ $\ds 437 \, 580$ $\ds$ $=$ $\ds \dfrac {935 \times 936} 2$ $\ds$ $=$ $\ds T_{935}$

 $\ds T_{2090} + T_{1869}$ $=$ $\ds 2 \, 185 \, 095 - 1 \, 747 \, 515$ $\ds$ $=$ $\ds 3 \, 932 \, 610$ $\ds$ $=$ $\ds \dfrac {2804 \times 2805} 2$ $\ds$ $=$ $\ds T_{2804}$

$\blacksquare$