Pythagorean Triples are sets of 3 positive integers where a² + b² = h². They are so called because the Pythagorean Theorem tells us that if the lengths of the sides of a triangle are proportional to the numbers in such a triple, then the triangle is a right triangle with "a" and "b" being the lengths of the legs and "h" being the length of the hypotenuse.

The JavaScript calculator below finds Pythagorean Triples either by increasing length of the shortest leg, or by increasing length of the hypotenuse. The algorithms for these two operations are different - it is not just a different sort, because that would be computationally inefficient. Euclid's Algorithm is used to eliminate triples (such as 6, 8, 10) which are simply multiples of others (3, 4, 5).

This calculator solves for Pythagorean Triples in which three integers have the relationship: a² + b² = h².

Euclid's Formula (circa 300 bc) generates all possible primitive Pythagorean Triples (not multiples of other triples) from two coprime numbers "m" and "n" if m > n and at least one of them is even. It may be necessary to swap "a" and "b".

a = m² - n²
b = 2mn
h = m² + n²

JavaScript uses IEEE double-precision floating point to represent numbers, and its accuracy is consistent with that representation. IEEE double precision floating point uses an exponent of 12 bits, a sign bit, and a mantissa of 51 bits. Thus integers up to about 2⁵¹ or 2 x 10¹⁵ can be represented without roundoff errors. This should be borne in mind when running integer calculations on very large numbers.

As with any web page written in JavaScript, you can see how I have coded the routines by doing a "View Source" from your browser.

Finding triples in order of the shortest leg.

To find triples in order of the shortest leg, we start with the beginning value entered and count upward, one at a time. For example, if we started with 3, we would search for all Triples that have 3 for their shortest leg, then for all with 4 as the shortest let, etc. etc.

Once we have chosen a trial value for "a", the first and smallest member of the triple, we need to consider how far up we need to search for "b", the second number of the triple. From the definition of a Pythagorean Triple we know that "a", "b", and "h" must all be integers. So for any given value of "a", we are looking for the lowest value of "b" for which a² is less than the difference between b² and (b + 1)².

This value is important. Failing to search values up to this limit will cause us to miss some triples. Searching values of "b" above this is a waste of time, since above this limit a² is smaller than the minimum required to reach the next square.

We can do a little algebra and find that (b+1)² = b² + 2b + 1. So when "b" becomes so large that 2b + 1 is larger than a², there is no further point in searching for larger values of "b", as a² is no longer large enough for the sum of the two squares to even reach (b + 1)².

So our limit for "b" is the next integer larger (to be safe) than a²/2.

It is important to note that after finding a value for "b" that produces a triple we cannot abandon the present value for "a", since there may be some larger value of "b" that combines with "a" to produce a larger "h". So we must continue on until the limit for "b" (as described above) has been reached. At that point we have finished with that value of "a" and can move on to a+1.

Example: If "a" = 20 then we must check all values of "b" up to a²/2 = 20²/2 = 200
20² + 21² = 29²
20² + 99² = 101²

An even better example:
60² + 91² = 109²
60² + 221² = 229²
60² + 899² = 901²

An illustration of the upper limit for "b", which in this case is a²/2 = 65²/2 = 2112.5
65² + 72² = 97²
65² + 2112² = 2113²

The algorithm continues chugging along until all values of "a" in the range requested have been searched. At that point it is time to output the triples that have been found. These are stored in global JavaScript variables as they are found, and printed out as the last step in either Column or Tab Delimited form from the stored variables.

Caution

If you have followed the previous argument you can see that in the above calculation, b can be as large as a²/2 and therefore in calculating "h", the hypotenuse, this upper value of "b" must be squared. This means that numbers as large as a⁴/4 can result. This implies that the upper limit of "a" is 9457 (because 9457⁴/4 = 1.9996 x 10¹⁵ which is near the upper range of integers that can be represented accurately by JavaScript). Therefore you should probably not use this calculator for numbers larger than that. Of course the practicalities of run time on real computers will also be a factor at that level [;-)

Finding triples in order of the shortest hypotenuse.

This is simpler and faster than finding triples associated with the lowest leg. One reason for this is that for any trial hypotenuse, defining the value of one leg defines value of the other. So the procedure is to choose the trial hypotenuse and step through values of the shorter leg. if the two legs are "a" and "b" and the hypotenuse is "h", then we step through the possible values for "a" and calculate what the corresponding value for "b" would be. For example:
a² + b² = h² and therfore b² = h² - a².

So for each candidate value of "a" we find the candidate value for "b". If the value of "b" is an integer, and it has no common factors with "a", we have found a triple.

Of course we have to check for all the values of "a" up to h/sqrt(2). It is not necessary to search above that, as once "a" goes above that, the value of "b" would be smaller than the value of "a" and we would just find duplicates.

Happy Computing!!

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