Perimeters and Pythagorean Triples

My diurnal age today is 26800 and one of the properties of this number is that it's a member of OEIS A010814: perimeters of integer-sided right triangles. Such numbers are relatively frequent (almost 22%) as can be seen in the data below:

5824th member is 26780

5825th member is 26784

5826th member is 26790

5827th member is 26796

5828th member is 26800

5829th member is 26808

5830th member is 26810

5831th member is 26814

5832th member is 26820

As I discovered however, it's not easy to find the sides of the triangles with these perimeters. The \(m,n\) method for generating primitive Pythagorean triples is shown in Figure 1:

Figure 1

Thus we have the shortest side \(a=m^2-n^2\) with \(m \gt n\) with \(b=2mn\) and the hypotenuse \(c=m^2+n^2\) where \(m\) and \(n\) are integers. Using these formulae in SageMathCell to generate the triples and associated perimeters, I found that the calculation timed out once the perimeters got even moderately large (far smaller than 26800 for example). Fortunately, there was a web resource that I could call upon. See Figure 2:

Figure 2: https://r-knott.surrey.ac.uk/Pythag/pythag.html

As can be seen, the calculator spits out a plethora of data including the perimeter. It turns out that there are three different right angled triangles with a perimeter of 26800: (6800, 8844, 11156), (5360, 10050, 11390) and (4355, 10800, 11645). If the triangle's dimensions are a multiple of those of a primitive triangle then this is shown as well. This website is certainly the ultimate resource for information about right-angled triangles.

Sometimes there is only one right-angled triangle with a given perimeter. For example, the one triangle with perimeter 26814 has dimensions (3052, 11685, 12077) and it is primitive.  Perimeters that arise from only one triangle form OEIS A098714: only one Pythagorean triangle of this perimeter exists. 26814 is the 2944th member of this sequence and so such numbers make up slightly under 11% of all numbers (at least in the range up to 26814).

The subsite referenced in Figure 2 is maintained by a Dr. Ron Knott and the main site contains links to many other interesting mathematical concepts. Here is some biographical information:

Ph.D (1980, University of Nottingham), M.Sc (1976, University of Nottingham), B.Sc (Pure Maths, University of Wales), C.Math, FIMA, C.Eng, MBCS, CITP

Visiting Fellow, Department of Mathematics,

formerly Lecturer in Mathematics and Computing Science Departments (1979-1998)

Faculty of Electronics and Physical Sciences,

University of Surrey,

Contact me initially by Email: ronknott at mac dot com

I was a lecturer in the Departments of Mathematics and Computing Science at the University of Surrey, Guildford, UK, for 19 years until September 1998 when I left to start working for myself making web pages for maths education sites.

I now give mathematics talks to students at schools and universities as well as to general audiences, teachers' conferences and Science Festivals on topics of the web pages above, especially the Fibonacci Numbers and why they occur so often in plants, Fun with Fractions, As Easy As Pi etc.

I now live in Bolton, near Manchester in NW England.