My previous post was written with the help of a few very useful tools:
The set of numbers can be partitioned into two subsets of the same size, such that the two sets have equal sums, sums of squares, sums of cubes, …, up to sums of th powers.
For example, for :
can be partitioned as
Amazingly, this can be done for any non-negative integer .
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 function 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.