PythagoreanTriples

A Pythagorean Triple is a list of three positive integers $(a,b,c)$ with the property that $a^2 + b^2 = c^2$. That is, they are integers that are the lengths of the two legs and the hypotenuse of a right triangle.

In this module, a PythagoreanTriple is a container that holds three such integers with the additional property that $a<b$.

Construction

Three-parameter construction

A Pythagorean triple can be constructed simply by specifying the lengths of the legs and hypotenuse of a right triangle

julia> using PythagoreanTriples

julia> T = PythagoreanTriple(4,3,5)
PythagoreanTriple(3, 4, 5)

If the three integers are not the side lengths of a right triangle, an error is thrown:

julia> T = PythagoreanTriple(4,5,6)
ERROR: (4, 5, 6) does not define a Pythagorean triple

In addition, the three side lengths can be given as a tuple:

julia> abc = (4,3,5)
(4, 3, 5)

julia> PythagoreanTriple(abc)
PythagoreanTriple(3, 4, 5)

Two-parameter construction

Pythagorean triples can be created from a pair of integers u,v in which the legs of the right triangle are abs(u^2 - v^2) and abs(2*u*v), and the hypotenuse is u^2 + v^2.

A user can give the values u and v to create a Pythagorean triple:

julia> T = PythagoreanTriple(2,5)
PythagoreanTriple(20, 21, 29)

Note that the integers u and v must be distinct and nonzero:

julia> PythagoreanTriple(0,5)
ERROR: (0, 5) does not yield a PythagoreanTriple

julia> PythagoreanTriple(2,2)
ERROR: (2, 2) does not yield a PythagoreanTriple

Generation

We call a Pythagorean triple (a,b,c) primitive provided gcd(a,b) == 1. Thus (3,4,5) is primitive but (6,8,10) is not.

Use TripleGenerator to create an iterator for primitive Pythagorean triples. The iterator TripleGenerator(n) will create n primitive triples:

julia> for t in TripleGenerator(5)
       println(t)
       end
PythagoreanTriple(3, 4, 5)
PythagoreanTriple(5, 12, 13)
PythagoreanTriple(8, 15, 17)
PythagoreanTriple(7, 24, 25)
PythagoreanTriple(20, 21, 29)

If n is omitted (or negative) the iterator will produce values endlessly. For example:

julia> for t in TripleGenerator()
           a,b,c = make_tuple(t)
           if a > 1000
               println(t)
               break
           end
       end
PythagoreanTriple(1007, 1224, 1585)

Note: The function is_primitive is available to check if a Pythagorean triple is primitive:

julia> t = PythagoreanTriple(3,4,5)
PythagoreanTriple(3, 4, 5)

julia> is_primitive(t)
true

julia> t = PythagoreanTriple(30,40,50)
PythagoreanTriple(30, 40, 50)

julia> is_primitive(t)
false

The function make_primitive takes a Pythagorean triple and returns a new triple having divided the three numbers by their greatest common divisor.

julia> t = PythagoreanTriple(6,8,10)
PythagoreanTriple(6, 8, 10)

julia> make_primitive(t)
PythagoreanTriple(3, 4, 5)

Scalar multiple

Pythagorean triples can be multiplied (on the left) by a positive integer. For example:

julia> p = PythagoreanTriple(5,12)
PythagoreanTriple(119, 120, 169)

julia> 10p
PythagoreanTriple(1190, 1200, 1690)

Finding parameters

All primitive Pythagorean triples (and some, but not all, non-primitive triples) can be constructed using two parameters p = PythagoreanTriple(u,v).

Given such a triple, the function get_parameters will return a u and v that creates that triple.

julia> p = PythagoreanTriple(5,12,13)
PythagoreanTriple(5, 12, 13)

julia> u,v = get_parameters(p)
(2, 3)

julia> PythagoreanTriple(u,v)
PythagoreanTriple(5, 12, 13)

While some non-primitive triples, such as (6,8,10), can be created this way, others, such as (30,40,50), cannot:

julia> p = get_parameters(6,8,10)
(1, 3)

julia> get_parameters(30,40,50)
ERROR: Unable to find parameters for (30, 40, 50) (non-primitive)

Extracting values

The function make_tuple returns a 3-tuple containing the three values:

julia> t = PythagoreanTriple(5,11)
PythagoreanTriple(96, 110, 146)

julia> make_tuple(t)
(96, 110, 146)

Sorting

Pythagorean triples can be compared using the usual <, <=, >, >= relations. Lists of triples can be sorted. They are sorted "reverse lexicographically". That is when we check if (a,b,c) is less than or equal to (aa,bb,cc) we first see if c ≤ cc. If so, then if b ≤ bb. And if so, then if a ≤ aa.

julia> list = collect(TripleGenerator(10))
10-element Vector{Any}:
 PythagoreanTriple(3, 4, 5)
 PythagoreanTriple(5, 12, 13)
 PythagoreanTriple(8, 15, 17)
 PythagoreanTriple(7, 24, 25)
 PythagoreanTriple(20, 21, 29)
 PythagoreanTriple(9, 40, 41)
 PythagoreanTriple(12, 35, 37)
 PythagoreanTriple(11, 60, 61)
 PythagoreanTriple(28, 45, 53)
 PythagoreanTriple(33, 56, 65)

julia> sort(list)
10-element Vector{Any}:
 PythagoreanTriple(3, 4, 5)
 PythagoreanTriple(5, 12, 13)
 PythagoreanTriple(8, 15, 17)
 PythagoreanTriple(7, 24, 25)
 PythagoreanTriple(20, 21, 29)
 PythagoreanTriple(12, 35, 37)
 PythagoreanTriple(9, 40, 41)
 PythagoreanTriple(28, 45, 53)
 PythagoreanTriple(11, 60, 61)
 PythagoreanTriple(33, 56, 65)

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