# Sequence of Triplets of Primitive Pythagorean Triangles with Same Area

## Sequence

The sequence of sets of $3$ primitive Pythagorean triangles which all have the same area begins:

Generator Lengths of sides Area
$\left({77, 38}\right)$ $\left({4485, 5852, 7373}\right)$ $13 \, 123 \, 110$
$\left({78, 55}\right)$ $\left({3059, 8580, 9109}\right)$ $13 \, 123 \, 110$
$\left({138, 5}\right)$ $\left({1380, 19 \, 019, 19 \, 069}\right)$ $13 \, 123 \, 110$
$\left({2035, 266}\right)$ $\left({4 \, 070 \, 469, 1 \, 082 \, 620, 4 \, 163 \, 285}\right)$ $2 \, 203 \, 385 \, 574 \, 390$
$\left({3306, 61}\right)$ $\left({10 \, 925 \, 915, 403 \, 332, 7 \, 309 \, 493}\right)$ $2 \, 203 \, 385 \, 574 \, 390$
$\left({3422, 55}\right)$ $\left({11 \, 707 \, 059, 376 \, 420, 11 \, 713, 109}\right)$ $2 \, 203 \, 385 \, 574 \, 390$
$\left({1610, 869}\right)$ $\left({1 \, 836 \, 939, 2 \, 798 \, 180, 3 \, 347 \, 261}\right)$ $2 \, 570 \, 042 \, 985 \, 510$
$\left({2002, 1817}\right)$ $\left({706 \, 515, 7 \, 275 \ , 268, 7 \, 309 \, 493}\right)$ $2 \, 570 \, 042 \, 985 \, 510$
$\left({2622, 143}\right)$ $\left({6 \, 854 \, 435, 749 \, 892, 6 \, 895 \, 333}\right)$ $2 \, 570 \, 042 \, 985 \, 510$
$\left({2201, 1166}\right)$ $\left({3 \, 484 \, 845, 5 \, 132 \, 732, 6 \, 203 \, 957}\right)$ $8 \, 943 \, 387 \, 723 \, 270$
$\left({2438, 2035}\right)$ $\left({1 \, 802 \, 619, 9 \, 922 \, 660, 10 \, 085 \, 069}\right)$ $8 \, 943 \, 387 \, 723 \, 270$
$\left({3565, 198}\right)$ $\left({12 \, 670 \, 021, 1 \, 411 \, 740, 12 \, 748 \, 429}\right)$ $8 \, 943 \, 387 \, 723 \, 270$
$\left({7238, 2465}\right)$ $\left({46 \, 312 \, 419, 35 \, 683 \, 340, 6 \, 203 \, 957}\right)$ $826 \, 290 \, 896 \, 699 \, 730$
$\left({9077, 1122}\right)$ $\left({81 \, 133 \, 045, 20 \, 368 \, 788, 10 \, 085 \, 069}\right)$ $826 \, 290 \, 896 \, 699 \, 730$
$\left({10 \, 434, 731}\right)$ $\left({108 \, 333 \, 995, 15 \, 254 \, 508, 12 \, 748 \, 429}\right)$ $826 \, 290 \, 896 \, 699 \, 730$
$\left({352 \, 538, 2 \, 999 \, 447}\right)$ $\left({8 \, 872 \, 399 \, 264 \, 365, 2 \, 114 \, 838 \, 092 \, 972, 9 \, 120 \, 965 \, 347 \, 253}\right)$ $9 \, 381 \, 843 \, 970 \, 167 \, 926 \, 138 \, 271 \, 390$
$\left({1 \, 931 \, 103, 2 \, 398 \, 838}\right)$ $\left({2 \, 025 \, 264 \, 953 \, 635, 9 \, 264 \, 806 \, 516 \, 628, 9 \, 483 \, 582 \, 546 \, 853}\right)$ $9 \, 381 \, 843 \, 970 \, 167 \, 926 \, 138 \, 271 \, 390$
$\left({3 \, 063 \, 347, 3 \, 215 \, 070}\right)$ $\left({952 \, 580 \, 262 \, 491, 19 \, 697 \, 750 \, 078 \, 580, 19 \, 720 \, 769 \, 947 \, 309}\right)$ $9 \, 381 \, 843 \, 970 \, 167 \, 926 \, 138 \, 271 \, 390$

These appear to be the only ones known.

## Historical Note

The first of these sets of $3$, with area $13 \, 123 \, 110$, was reported by Martin Gardner as having been discovered by Charles L. Shedd in $1945$.

The following $3$ sets of $3$ were discovered by Randall L. Rathbun in $1986$.

The $5$th set of $3$ was discovered by Dan Hoey and Randall L. Rathbun in $1990$.

The $6$th set of $3$ was discovered by Duncan Moore using an exhaustive computer search over areas, and reported in The Math Forum on the $2$nd of March, $2017$. The search took roughly $35$ GHz-days.