The Perfect Right Triangle and Quantum Coordinates in Cartesian Space

In high school geometry, we learned of the perfect right triangle. Both sides and the hypotenuse are integers (whole numbers). The perfect right triangle shown was 3, 4, 5. (3 sq + 4 sq = 9 + 16 = 25 = 5 sq). I wondered if there were others.

Years later, I seriously searched for other perfect right triangles. I began with a list of the squares of whole numbers and the difference between one square and the next (the "delta").

Discovery #1: The series of deltas = the series of odd numbers. I looked for deltas which are squares. Since the series of deltas is the series of odd numbers, the square of every odd number is the difference between two squares.

Discovery #2: Every odd number 3 and above is the side of a right triangle. Implication of Discovery # 2: Since there is an infinite number of odd numbers, there is an infinite number of perfect right triangles. Next, a formula for finding the other sides and hypotenuse of a right triangle was worked out. There were two formulas, one for odd numbers, one for even. It turned out that no matter what the number was, odd or even, there were two other numbers for a perfect right triangle.

Discovery #3: Every whole number (integer) is a side of a perfect right triangle. Considering a right triangle as half of a 2 dimensional box cut diagonally, formulas were worked out for 3 dimensional boxes, then for 4 dimensional, etc. Again, there is at least one box for every number in every dimension.

Discovery #4: A perfect right triangle has one odd side, one even side and an odd hypotenuse. If the hypotenuse is even, both the sides are even, and each of the three lengths is the same multiple of an odd-even-odd perfect right triangle, which is the root (or prime) right triangle. The hypotenuse of a prime perfect right triangle is always the longer side + 1 if the shorter side is odd, and the longer side + 2 if the shorter side is even. There are multiples of these patterns.

Discovery #5: There are an unlimited number of perfect right triangles (or perfect boxes) in an unlimited number of dimensions. Charting these numbers on Cartesian grids yields discrete (aka "quantum") points (coordinates), since these are all integral numbers. Therefore I call them "Quantum Coordinates in Cartesian Space." Their number is infinite.

Here is a sample of perfect right triangles calculated from the formulas:

For Side A is Odd: (A sq - 1)/2 = B, (A sq + 1)/2 = C
A, B, C
1, 0, 1
3, 4, 5
5, 12, 13
7, 24, 25
9, 40, 41
11, 60, 61
13, 84, 85
15, 112, 113
17, 144, 145
19, 180, 181

For Side A is Even: A sq / 4 - 1 = B, A sq / 4 + 1 = C
A, B, C
2, 0, 2
4, 3, 5
6, 8, 10
8, 15, 17
10, 24, 26
12, 35, 37
14, 48, 50
16, 63, 65
18, 80, 82
20, 99, 101


“Every odd number above 3 is the hypotenuse of a right triangle.” Neither 7 nor 11 is the length of the hypotenuse of a right triangle with sides whose lengths are integers. “Every whole number is a side of a perfect right triangle.” The shortest possible length of a side of a right triangle, all of whose sides are integers, is 3, and the 3,4,5 triangle is the only one that has a side of length 3. Perhaps I don’t understand what you mean by ‘perfect right triangle’.
I reviewed and revised the article shortly after seeing Rudy L's comment.
LocoMotives has found other Perfect Right Triangles besides those in the formulae listed. Analysis of these may reveal new classes with new formulae.


Wesley G. V.

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