Perimeters of Pythagorean Right-Angled Triangles

Right-angled triangles with whole number sides have fascinated mathematicians and number enthusiasts since well before 300 BC when Pythagoras wrote about his famous "theorem". The oldest mathematical document in the world, a little slab of clay that would fit in your hand, can be seen a list of such triangles. So what is so fascinating about them? This page starts from scratch and has lots of facts and figures with several online calculators to help with your own investigations. Source.

This site from which this quote was taken is a great resource and I was prompted to visit it after investigating the number 26322 that constituted my diurnal age today (Tuesday, April 27th 2021). One of the properties of this number is that its a member of OEIS A098714:

A098714

Only one Pythagorean triangle of this perimeter exists.

The members of this sequence are relatively numerous. 26322 is the 2888th member of this sequence so they comprise 10.97% of all numbers up to 26322, a slightly higher frequency than the primes and lucky numbers. However, if the restriction that only one triangle can exist is removed, then 26322 is the 5720th member of OEIS A010814 and so 21.73% of all numbers up to this point are perimeters of one or more Pythagorean triangles.

A010814

Perimeters of integer-sided right triangles.

Let's return to the Pythagorean triangle of perimeter 26322 (units) for a moment and determine what the lengths of the sides of the triangle are that make up this perimeter. Figure 1 shows the SageMath code that I developed for this purpose with permalink attached.

Figure 1: permalink

The output from this code reveals that the sides are 3424, 11193 and 11705 (units). Figure 2 shows a not necessarily to scale representation of the triangle.

Figure 2

Now let's return to the site mentioned at the start of this post and see what interesting information can be extracted about perimeters of Pythagorean right-angled triangles. Figure 3 shows a table listing all Pythagorean Triples with sides up to 100 arranged in order of hypotenuse (longest side).

Figure 3

Primitive Pythagorean triads are those whose greatest common divisor (gcd) is 1 e.g. 3, 4, 5. The triad 6, 8, 10 on the other hand is not primitive because the gcd is 2. The site lists some other interesting sequences from the OEIS that follow on from those mentioned earlier.

A099831

Perimeters of Pythagorean triangles that can be constructed in exactly two different ways.

The members of this sequence are far less frequent than those for triangles that can be constructed in only one way. Initial members of OEIS A099831 are:

60, 84, 90, 132, 144, 210, 264, 270, 288, 300, 312, 330, 390, 408, 432, 440, 450, 456, 462, 468, 510, 520, 546, 552, 570, 576, 588, 612, 616, 680, 684, 690, 700, 728, 760, 770, 800, 810, 816, 828, 870, 910, 912, 918, 920, 952, 1044, 1064, 1100, 1104, 1116, ...

A099832

Perimeters of Pythagorean triangles that can be constructed in exactly three different ways.

The initial members of OEIS A099832 are:

120, 168, 180, 252, 280, 336, 396, 528, 540, 560, 600, 624, 792, 864, 880, 936, 1040, 1050, 1056, 1120, 1176, 1224, 1232, 1248, 1350, 1368, 1380, 1404, 1456, 1620, 1632, 1650, 1656, 1710, 1728, 1740, 1760, 1764, 1824, 1836, 1860, 1960, 2002, 2052, 2080, ...

A099833

Perimeters of Pythagorean triangles that can be constructed in exactly four different ways.

The initial members of OEIS A099833 are:

240, 360, 480, 504, 630, 672, 756, 780, 900, 960, 990, 1020, 1092, 1140, 1170, 1188, 1344, 1386, 1400, 1428, 1530, 1540, 1596, 1638, 1820, 1920, 1932, 1950, 2070, 2112, 2240, 2244, 2268, 2376, 2380, 2448, 2496, 2508, 2610, 2652, 2660, 2688, 2736, 2800, ...

A156687

Perimeters of Pythagorean triangles that can be constructed in exactly five different ways.

 The initial members of OEIS A156687 are:

420, 660, 924, 1008, 1080, 1200, 1512, 1584, 1716, 1800, 1872, 1890, 2700, 3150, 3168, 3240, 3480, 3528, 3570, 3720, 3744, 4410, 4440, 4536, 4590, 4704, 4872, 4896, 4950, 5208, 5292, 5472, 5600, 5670, 6000, 6090, 6210, 6216, 6624, 6630, 6660, 6888, ...

With all members of these sequences, the SageMath algorithm can be applied to determine what the sides of the triangles are. For example, the first member of OEIS A156687, 420, yields:

• 105, 140, 175
• 70, 168, 182
• 120, 126, 174
• 60, 175, 185
• 28, 195, 197

As can be seen, of the above five triads, only one is primitive (28, 195, 197). It is interesting to consider only primitive triads and in doing so, some interesting results emerge:

• The first primitive Pythagorean triangles with the same perimeter are 195, 748, 773 and 364, 627, 725 with a perimeter of 1716. The next such perimeters are 2652, 3876, 3960, ... and these perimeters constitute OEIS A024408.

A024408

Perimeters of more than one primitive Pythagorean triangle.

We have to go up to a perimeter of 14280 before we find three primitive triads with the same perimeter: 119, 7080, 7081 and 168, 7055, 7057 and 3255, 5032, 5993. It should be noted that there are 19 distinct triads with a perimeter of 14280 but only the three mentioned are primitive. Similarly the next member of the sequence 72930 has 16 triads but only three that are primitive (2992, 34905, 35033 and 7905, 32032, 32993 and 18480, 24089, 30361). The initial members of OEIS A024408 are:

1716, 2652, 3876, 3960, 4290, 5244, 5700, 5720, 6900, 6930, 8004, 8700, 9300, 9690, 10010, 10788, 11088, 12180, 12876, 12920, 13020, 13764, 14280, 15252, 15470, 15540, 15960, 16380, 17220, 17480, 18018, 18060, 18088, 18204, 19092, 19320, 20592, 20868, ...

It can be noted that any right-angled triangle whose side lengths are a Pythagorean triple is a Heronian triangle, as the side lengths of such a triangle are integers, and its area is also an integer, being half of the product of the two shorter sides of the triangle, at least one of which must be even. See earlier post on Heronian triangles.