This is a link to a historical project of mine, hosted on Albert Gräf’s project page.
There is full documentation available.
In a recent post I described a method of generating the simplest primitive Pythagorean triple (a,b,c) where one of the angles of the triangle with sides a, b and c is θ° to within some (small) error bound Δθ°.
One of the steps was, given the cosine C of the angle [from step (2)]:
(3) Calculate the Farey ratio approximant …
Now, that function
seemed semi-familiar, resembling functions that occur in trigonometric or hyperbolic identities.
An example is:
A little further investigation, and reading around, including the Wikipedia articles on trigonometric identities, and in particular on those of the tangent half-angle, revealed that the Farey ratio approximant does in fact correspond directly to a simple trigonometric funtion of the angle:
The slightly simplified method follows.
|θ° ± 0.01°||a||b||c||θ°||±%ε|
A longer table follows
The next post provides a worked example of the method.
ADDENDUM [20-10-2015]: A slight simplification of the method below is described in an addendum post.
Suppose you wish to find the simplest primitive Pythagorean triple (a,b,c) where one of the angles of the triangle with sides a, b and c is θ° to within some (small) error bound Δθ°.
A Pythagorean triple (a,b,c) is such that:
a,b,c ∈ ℕ₀ (i.e. are natural numbers ≥ 0), and
a² + b² = c²
A primitive Pythagorean triple (a,b,c) is one such that also
a ⊥ b (i.e. a and b are coprime, i.e. have no common factors),
a ⊥ c, and
b ⊥ c
that is, a, b and c are pairwise coprime.
The method follows.
In an earlier post, I showed the finite version of this result:
This might be seen to hold as both sums are
Whilst doodling with the Fibonacci sequence
I found some interesting formulae:
The first of these is not new, but I did not find the other two on the web.
where ω is a primitive cube root of 1 given by
are often represented in the form
(noting that )
and perhaps abbreviated by pair notation such as
(where the asterisk is to distinguish this notation from that introduced below)
Here is a more intuitive representation that is simpler to manipulate and reason about.