Cite this DOI

Hilal Boyraz (2026). New Pythagorean Triangles from Triangular Numbers. *Journal of Science & Technology*, *11*(01), 01. https://doi.org/10.46243/jst.2026.v11.i01.pp01-055

@article{hilalboyraz2026pythagorean,
  author    = {Hilal Boyraz},
  title     = {{New Pythagorean Triangles from Triangular Numbers}},
  journal   = {Journal of Science \& Technology},
  year      = {2026},
  month     = {jan},
  volume    = {11},
  number    = {01},
  pages     = {01},
  publisher = {Longman Publishers},
  issn      = {2456-5660},
  doi       = {10.46243/jst.2026.v11.i01.pp01-055},
  url       = {https://doi.org/10.46243/jst.2026.v11.i01.pp01-055},
  language  = {en},
  abstract  = {Special right triangles formed from special numbers are known. 3-4-5, 5-12-13 are some examples. The study began by working with triangular numbers and establishing a relationship between the Pythagorean theorem and triangular numbers. The aim of the study is to obtain special numbers that form special right triangles using the Pythagorean theorem and to generalize these numbers. The problem statement, "Can special right triangles be formed from the terms of triangular numbers using the Pythagorean theorem?" was addressed. The squares of every two consecutive terms of the triangular numbers were taken, and the difference between the squares was calculated. When the square root of these differences was taken, it was observed that the difference between the two consecutive terms could be written as a multiple of the square root itself. Using the Pythagorean theorem, a special triangle is formed where the smaller of the consecutive terms is one of the legs, the larger one is the hypotenuse, and the other leg is a multiple of the square root of their difference. The process of generating special right triangles was performed using a Python AI program. It was found to be very easy to perform these operations in Python. By continuing this process, special numbers that form special triangles and a general rule for these special right triangles were obtained by utilizing the general rule for triangular numbers. This general rule showed that it is easier to obtain special triangles from consecutive terms of triangular numbers. Our study is significant in terms of obtaining special numbers in a Python program using the Pythagorean theorem for the differences between the squares of triangular numbers, and generalizing the special right triangles formed from these special numbers.}
}

TY  - JOUR
TI  - New Pythagorean Triangles from Triangular Numbers
AU  - Hilal Boyraz
JO  - Journal of Science & Technology
PY  - 2026
DA  - 2026/01/30/
VL  - 11
IS  - 01
SP  - 01
PB  - Longman Publishers
SN  - 2456-5660
LA  - en
AB  - Special right triangles formed from special numbers are known. 3-4-5, 5-12-13 are some examples. The study began by working with triangular numbers and establishing a relationship between the Pythagorean theorem and triangular numbers. The aim of the study is to obtain special numbers that form special right triangles using the Pythagorean theorem and to generalize these numbers. The problem statement, "Can special right triangles be formed from the terms of triangular numbers using the Pythagorean theorem?" was addressed. The squares of every two consecutive terms of the triangular numbers were taken, and the difference between the squares was calculated. When the square root of these differences was taken, it was observed that the difference between the two consecutive terms could be written as a multiple of the square root itself. Using the Pythagorean theorem, a special triangle is formed where the smaller of the consecutive terms is one of the legs, the larger one is the hypotenuse, and the other leg is a multiple of the square root of their difference. The process of generating special right triangles was performed using a Python AI program. It was found to be very easy to perform these operations in Python. By continuing this process, special numbers that form special triangles and a general rule for these special right triangles were obtained by utilizing the general rule for triangular numbers. This general rule showed that it is easier to obtain special triangles from consecutive terms of triangular numbers. Our study is significant in terms of obtaining special numbers in a Python program using the Pythagorean theorem for the differences between the squares of triangular numbers, and generalizing the special right triangles formed from these special numbers.
DO  - 10.46243/jst.2026.v11.i01.pp01-055
UR  - https://doi.org/10.46243/jst.2026.v11.i01.pp01-055
ER  -

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    "DOI": "10.46243/jst.2026.v11.i01.pp01-055",
    "URL": "https://doi.org/10.46243/jst.2026.v11.i01.pp01-055",
    "title": "New Pythagorean Triangles from Triangular Numbers",
    "source": "Smart Scholars DOI Registry",
    "container-title": "Journal of Science & Technology",
    "author": [
        {
            "family": "Hilal Boyraz"
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    "issued": {
        "date-parts": [
            [
                2026,
                1,
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    },
    "volume": "11",
    "issue": "01",
    "page": "01",
    "publisher": "Longman Publishers",
    "language": "en",
    "abstract": "Special right triangles formed from special numbers are known. 3-4-5, 5-12-13 are some examples. The study began by working with triangular numbers and establishing a relationship between the Pythagorean theorem and triangular numbers. The aim of the study is to obtain special numbers that form special right triangles using the Pythagorean theorem and to generalize these numbers. The problem statement, \"Can special right triangles be formed from the terms of triangular numbers using the Pythagorean theorem?\" was addressed. The squares of every two consecutive terms of the triangular numbers were taken, and the difference between the squares was calculated. When the square root of these differences was taken, it was observed that the difference between the two consecutive terms could be written as a multiple of the square root itself. Using the Pythagorean theorem, a special triangle is formed where the smaller of the consecutive terms is one of the legs, the larger one is the hypotenuse, and the other leg is a multiple of the square root of their difference. The process of generating special right triangles was performed using a Python AI program. It was found to be very easy to perform these operations in Python. By continuing this process, special numbers that form special triangles and a general rule for these special right triangles were obtained by utilizing the general rule for triangular numbers. This general rule showed that it is easier to obtain special triangles from consecutive terms of triangular numbers. Our study is significant in terms of obtaining special numbers in a Python program using the Pythagorean theorem for the differences between the squares of triangular numbers, and generalizing the special right triangles formed from these special numbers.",
    "ISSN": "2456-5660"
}

From the record as registered (version 2) — the record and its history. Programs: https://registry.smartscholars.in/api.php?action=cite&doi=10.46243%2Fjst.2026.v11.i01.pp01-055 gives all four in one JSON answer.