My previous post was written with the help of a few very useful tools:

## Tools for Writing Mathematical Blog Posts

Wednesday, 9 March 2016## The Partition Sum of Powers Theorem

Tuesday, 8 March 2016The 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

so that

and, lastly,

Amazingly, this can be done for any non-negative integer .

## QiX

Tuesday, 24 November 2015This is a link to a historical project of mine, hosted on Albert Gräf’s project page.

QiX is a library for Albert Gräf’s Q programming language adding support for univariate polynomials.

There is full documentation available.

## Q+Q

Tuesday, 24 November 2015This is a link to a historical project of mine, hosted on Albert Gräf’s project page.

Q+Q is a library for Albert Gräf’s Q programming language adding support for the rational numbers, ℚ.

There is full documentation available.

## Generating Approximate Pythagorean Angles (ADDENDUM) – Simplified Method

Tuesday, 20 October 2015In 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.

## Generating Approximate Pythagorean Angles (IV) – Derivation and Proof of The Method

Saturday, 10 October 2015In the previous post is a table of values.

Suppose you wish to find the simplest primitive Pythagorean triangle (a,b,c) where one of the angles is θ° to within some (small) error bound Δθ°.

Here’s the derivation of the method which was given in an earlier post.

## Generating Approximate Pythagorean Angles (III) – A Table for (1/100)°

Saturday, 10 October 2015The previous post provides a worked example of the method.

The next post provides a derivation and proof of the method.

### Short Table

θ° ± 0.01° |
a |
b |
c |
θ° |
±%ε |

5° | 33425 | 2928 | 33553 | 5.006° | 62.8% |

10° | 6351 | 1120 | 6449 | 10.001° | 12.9% |

15° | 1419 | 380 | 1469 | 14.992° | −82.8% |

20° | 66005 | 24012 | 70237 | 19.991° | −90.9% |

25° | 16272 | 7585 | 17953 | 24.992° | −79.8% |

30° | 2911 | 1680 | 3361 | 29.990° | −98.4% |

35° | 7623 | 5336 | 9305 | 34.992° | −84.8% |

40° | 20424 | 17143 | 26665 | 40.009° | 86.1% |

45° | 4059 | 4060 | 5741 | 45.007° | 70.6% |

A longer table follows

## Generating Approximate Pythagorean Angles (II) – A Worked Example

Saturday, 10 October 2015The previous post describes the method.

In the next post is a table of values.

As an example, let

θ = 24°

Δθ = 0.001°

## Generating Approximate Pythagorean Angles (I) – The Method

Saturday, 10 October 2015The 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.

## Reversal of Indices in Infinite Triangular Sums

Friday, 31 January 2014We have:

because

In an earlier post, I showed the finite version of this result:

This might be seen to hold as both sums are