Pythagorean Triples Calculator

The Pythagorean triple is a number formed from three consecutive integers: 1, 2 and 3. There are five possible combinations of numbers that form the Pythagorean triple. These combinations are 1, 2, 4, 8 and 16.

1, 2, 4, 8 and 16 are the only possible combinations that form the Pythagorean triple. All other combinations have no mathematical meaning. They do not even exist in nature!

There are two ways to determine if a number is in the correct combination for forming the Pythagorean triple. One way is to use the Pythagorean theorem which states that any pair of numbers cannot be divided into smaller sets than those pairs. For example, (2 + 6) / 7 = 0 and (4 – 9) / 10 = 0.

How many different ways are there to form the Pythagorean triple?

Let’s say you want to figure out which combination of numbers forms the Pythagorean triple. You could try all possible combinations and see if any of them produce a result. Or, you could use a computer program called a pythagorean tricorder (or pthtr). A pthtr is like an astrolabe or a sextant but it uses numbers instead of stars or planets.

You should start off by putting either the largest or smallest number of the combination is first. This is called the base. For example, if the combination is 2, 3, 5 then you put 2 as the base. If you put 3 as the base then the pthtr shows you that the combination is not a Pythagorean triple.

The remaining part of the combination becomes the length. In this case, it’s 5. Now, press the buttons on the pthtr in this order:

You see a countdown from 3 to 0. When it reaches 0, you must press the Show Answer button. If the combination is a true pythagorean triple then the pthtr will show you some information about it. In this case, you see that 2, 3 and 5 (or 2, 5 and 10 if you use the 10-based system) are indeed a pythagorean triple!

So, you enter 2 as the base and 5 as the length. This shows that 22 + 32 = 5, which is true.

The pthtr then shows you that 52 = 25, which is true as well. It takes the positive square root of each side of the equation. This gives you 5 as the first part of your answer. You need to do one more check to see if this generated a valid answer.

The final check uses 3 as the base.

After you have seen this information, the pthtr has something more to show you. What is it?

Pythagorean triples appear all over the place in nature and in math. There are 5 that are most common. These are the 5 that you will probably see most often. They are: 1, 2, 4; 1, 3, 5; 2, 3, 5; 3, 4, 5 and 5, 12, 13.

How many different ways are there to form the pythagorean triple?

You have now successfully used a pythagorean tricorder! You can now see that there are an infinite amount of ways to form the pythagorean triple. Some of these ways are more likely than others but they all exist!

You wonder if the pthtr is able to find the 5 most common ways to form a pythagorean triple. You give it a try and…

The pthtr shows you that there are 5 different ways to form a pythagorean triple. These are the 5 that show up most often.

The pythagorean tricorder is a fictional device seen in the Star Trek TV series. It is used to solve math problems. It was first used in an episode called, The Apple. A space craft that visited Earth in the past left the tri corder on it.

The tri corder helps Captain Kirk to solve the mysteries of a malfunctioning probe and to escape from a barren planet.

The pthtr consists of 3 lights; red, yellow and green. The red light means that the numbers that you entered are incorrect. If the yellow light turns on then you have entered a true pythagorean triple. The green light means that you have found one of the 5 most common pythagorean triples.

This article shows you how to build your own pythagorean tricorder. What is a pythagorean triple? It is a combination of 2 numbers that are related to each other through Pythagoras’ Theorem. For example, the pythagorean triple (3, 4, 5) satisfies the equation 32 + 42 = 52. The pthtr shows you that there are an infinite number of pythagorean triples and most of them don’t appear anywhere in nature or in math problems.

How does the pthtr know if a combination is a true pythagorean triple? It checks it against one specific pythagorean triple, (3, 4, 5). The pthtr does this by multiplying the length of the sides of the triangle. This generates a true pythagorean triple if and only if the generated triple is equal to the one that is stored inside the pthtr.

The pthtr is also able to find one of the 5 most common pythagorean triples using a similar check. It stores these 5 pythagorean triples inside of it.

__Sources & references used in this article:__

- Solving and verifying the boolean pythagorean triples problem via cube-and-conquer (MJH Heule, O Kullmann, VW Marek – International Conference on Theory …, 2016 – Springer)

https://link.springer.com/chapter/10.1007/978-3-319-40970-2_15 - Datasets on the statistical and algebraic properties of primitive Pythagorean triples (HI Okagbue, MO Adamu… – Data …, 2017 – eprints.covenantuniversity.edu.ng)

http://eprints.covenantuniversity.edu.ng/10516/ - Methods and traditions of Babylonian mathematics: Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations (J Friberg – Historia Mathematica, 1981 – Elsevier)

https://www.sciencedirect.com/science/article/pii/0315086081900690 - Pythagoras Triples Explained via Central Squares. (LT Gomes – Australian Senior Mathematics Journal, 2015 – ERIC)

https://eric.ed.gov/?id=EJ1093370 - Generalizations of classical results on Jeśmanowicz’conjecture concerning Pythagorean triples II (J Valley, CM Tutors – 2007)

https://www.sciencedirect.com/science/article/pii/S0022314X14000493 - Primitive pythagorean triples (T Miyazaki, P Yuan, D Wu – Journal of Number Theory, 2014 – Elsevier)

https://www.tandfonline.com/doi/pdf/10.1080/07468342.1992.11973493