The Pythagorean theorem is a well known result about properties of right triangles.

Pythagoras lived in Greece about 500 B.C. His most well-known result is the Pythagorean theorem. We are not otherwise concerned about Pythagoras and will ignore the fact that the result named for him appears to have been known well before his time and in many other places on the earth.

Start with a right triangle.

In the image, the (red) vertical/rise y distance is 3, the (green) horizontal/run x distance is 4, and the (blue) hypotenuse distance is 5.

A computer program can be used to find Pythagorean triples.

Here is a simple and somewhat clear Lua program, not necessarily the most efficient, to find all Pythagorean triples whose sides a and b are from 1 to 32.

Here is the Lua code [#1]
local n1 = 32 local list1 = {} for a1=1,n1 do for b1=a1,n1 do local c2 = a1*a1+b1*b1 local c1 = math.floor(math.sqrt(c2)+0.5) if (c1*c1) == c2 then table.insert(list1,{c1, a1, b1}) end end end table.sort(list1, function(a, b) return a[1] < b[1] end) for _,item1 in ipairs(list1) do local c1 = item1[1] local a1 = item1[2] local b1 = item1[3] print(string.format("%d = %d^2 = %d^2 + %d^2 = %d + %d",c1*c1,c1,a1,b1,a1*a1,b1*b1)) end

Here is the output of the Lua code.
25 = 5^2 = 3^2 + 4^2 = 9 + 16 100 = 10^2 = 6^2 + 8^2 = 36 + 64 169 = 13^2 = 5^2 + 12^2 = 25 + 144 225 = 15^2 = 9^2 + 12^2 = 81 + 144 289 = 17^2 = 8^2 + 15^2 = 64 + 225 400 = 20^2 = 12^2 + 16^2 = 144 + 256 625 = 25^2 = 15^2 + 20^2 = 225 + 400 625 = 25^2 = 7^2 + 24^2 = 49 + 576 676 = 26^2 = 10^2 + 24^2 = 100 + 576 841 = 29^2 = 20^2 + 21^2 = 400 + 441 900 = 30^2 = 18^2 + 24^2 = 324 + 576 1156 = 34^2 = 16^2 + 30^2 = 256 + 900 1225 = 35^2 = 21^2 + 28^2 = 441 + 784 1600 = 40^2 = 24^2 + 32^2 = 576 + 1024
Note that the commonly used example values 3, 4, and 5 is the minimal example of a Pythagorean triple.

Note that to determine if three integer values form a Pythagorean triple, it is not necessary to do a square root operation. (left as an exercise)