Triangular Number Pairs with Triangular Sum and Difference

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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

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