HOW MANY HERONIAN TRIANGLES HAVE A GIVEN LONGEST SIDE? A COMPLETE COUNT UP TO 2000 AND THE RIGHT TRIANGLES HIDDEN INSIDE

Abstract

A Heronian triangle has integer sides and an integer area. Sequence A054875 of the On-Line Encyclopedia of Integer Sequences (OEIS) gives the number a(n) of Heronian triangles whose longest side is exactly n, but its entry has no formula and no references. I wrote down three questions (for which n is a(n) = 0, what share of the triangles is right-angled, and what share is glued from two integer right triangles along an altitude) and made a forecast: about 300 triangles with longest side at most 100. All 668,167,500 integer triangles with longest side at most 2000 were then checked with Heron's formula, and a second enumeration by a different parametrisation gave the same list. There are 40,764 Heronian triangles with longest side at most 2000, and the counts agree term by term with the OEIS tables. The forecast was close: there are 321 such triangles up to 100. The total grows faster than the side itself (the average of a(n) is 4.6 for 50 < n ≤ 100 and 26.7 for 1000 < n ≤ 2000), but a(n) = 0 for 333 values of n ≤ 2000. None of these n has a prime factor of the form 4k + 1, and the paper proves that a(n) = 0 whenever n is a power of a prime p ≡ 3 (mod 4) (for primes this was recorded in the OEIS as an observation) or n = 2p for such a prime p ≥ 7; these two families give 254 of the 333 zeros. Right triangles make up only 4.9% of the 40,764, and their share falls across the range. Of all triangles, 32.6% are glued from two integer right triangles, 30.6% are the difference of two such triangles, and 32.0% have no integer altitude at all; for 1000 < n ≤ 2000 this last group is already the largest. Finally, as also follows from earlier work [3, 4], every altitude of a Heronian triangle is a fraction whose denominator has only prime factors of the form 4k + 1.

Keywords: Heronian triangle, Pythagorean triple, integer sequence, OEIS A054875, Diophantine equation, enumeration

References

1. Carlson J.R. Determination of Heronian triangles // The Fibonacci Quarterly. 1970. Vol. 8, No. 5. P. 499-506, 551. URL: https://www.fq.math.ca/Scanned/8-5/carlson-a.pdf (accessed: 11.10.2026).
2. Kurz S. On the generation of Heronian triangles // Serdica Journal of Computing. 2008. Vol. 2, No. 2. P. 181-196. DOI: 10.55630/sjc.2008.2.181-196.
3. Stănică P., Sarkar S., Sen Gupta S., Maitra S., Kar N. Counting Heron triangles with constraints // Integers. 2013. Vol. 13. Paper A3. 17 p. URL: https://math.colgate.edu/~integers/n3/n3.pdf (accessed: 11.10.2026).
4. Yiu P. Heron triangles which cannot be decomposed into two integer right triangles: talk at the 41st Meeting of the Florida Section of the Mathematical Association of America, Florida Southern College, Lakeland, 15-16 February 2008. 47 p. URL: https://web.archive.org/web/20161230000255/http://math.fau.edu/yiu/Southern080216.pdf (accessed: 11.10.2026).
5. Sequence A054875: Number of pairwise incongruent triangles with integer sides and positive integer area and longest side of length n // The On-Line Encyclopedia of Integer Sequences. OEIS Foundation Inc. URL: https://oeis.org/A054875 (accessed: 11.10.2026).
6. Sequences A054876, A055592, A055593, A055594, A055595, A096467, A239246 and A322105 // The On-Line Encyclopedia of Integer Sequences. OEIS Foundation Inc. URL: https://oeis.org (accessed: 12.10.2026).
7. Sequence A227166: Areas of indecomposable non-Pythagorean primitive integer Heronian triangles, sorted increasingly; Sequence A239978: Areas of indecomposable primitive integer Heronian triangles (including primitive Pythagorean triangles), in increasing order // The On-Line Encyclopedia of Integer Sequences. OEIS Foundation Inc. URL: https://oeis.org/A227166, https://oeis.org/A239978 (accessed: 11.10.2026).
8. Dinesh Kumar A., Vasuki M. A study on Pythagorean triples // International Journal of Interdisciplinary Research in Arts and Humanities. 2016. Vol. 1, Issue 1. P. 14-21. URL: https://ijirah.dvpublication.com/uploads/667650e07c1fe_3.pdf (accessed: 11.10.2026).
9. Stein W. Elementary Number Theory: Primes, Congruences, and Secrets. Version of 23 January 2017. 172 p. URL: https://wstein.org/ent/ent.pdf (accessed: 12.10.2026).

How to cite

Kenesbek M. HOW MANY HERONIAN TRIANGLES HAVE A GIVEN LONGEST SIDE? A COMPLETE COUNT UP TO 2000 AND THE RIGHT TRIANGLES HIDDEN INSIDE // Central Asian Scientific Journal. – 2026. – No. 4(32), Vol. 1. – Pp. 263–273.