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 $

$=$

$\ds T_{2804}$

$\blacksquare$

Sources

1919: Leonard Eugene Dickson: History of the Theory of Numbers: Volume $\text { II }$

1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.3$ Early Number Theory: Problems $1.3$: $5 \ \text {(a)}$
1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $15$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $15$