Editing Pythagorean triple (section)

===Almost-isosceles Pythagorean triples===

No Pythagorean triples are [[isosceles]], because the ratio of the hypotenuse to either other side is {{radic|2}}, but [[Square root of 2#Proofs of irrationality|{{radic|2}} cannot be expressed as the ratio of 2 integers]].

There are, however, [[Special right triangles#Almost-isosceles Pythagorean triples|right-angled triangles]] with integral sides for which the lengths of the [[Cathetus|non-hypotenuse sides]] differ by one, such as,

:<math>3^2+4^2 = 5^2</math>

:<math>20^2+21^2 = 29^2</math>

and an infinite number of others. They can be completely parameterized as,

:<math>\left(\tfrac{x-1}{2}\right)^2+\left(\tfrac{x+1}{2}\right)^2 = y^2</math>

where {''x, y''} are the solutions to the [[Pell equation]] <math>x^2-2y^2 = -1.</math>

If {{math|''a''}}, {{math|''b''}}, {{math|''c''}} are the sides of this type of primitive Pythagorean triple then the solution to the Pell equation is given by the [[Recurrence relation|recursive formula]]

:<math>a_n=6a_{n-1}-a_{n-2}+2</math> with <math>a_1=3</math> and <math>a_2=20</math>
:<math>b_n=6b_{n-1}-b_{n-2}-2</math> with <math>b_1=4</math> and <math>b_2=21</math>
:<math>c_n=6c_{n-1}-c_{n-2}</math> with <math>c_1=5</math> and {{tmath|1=c_2=29}}.<ref>{{cite OEIS |A001652 |mode=cs2}}; {{cite OEIS |A001653 |mode=cs2}}</ref>

This sequence of primitive Pythagorean triples forms the central stem (trunk) of the [[Tree of primitive Pythagorean triples|rooted ternary tree]] of primitive Pythagorean triples.

When it is the longer non-hypotenuse side and hypotenuse that differ by one, such as in

:<math>5^2+12^2 = 13^2</math>

:<math>7^2+24^2 = 25^2</math>

then the complete solution for the primitive Pythagorean triple {{math|''a''}}, {{math|''b''}}, {{math|''c''}} is

:<math>a=2m+1, \quad b=2m^2+2m, \quad c=2m^2+2m+1</math>

and

:<math>(2m+1)^2+(2m^2+2m)^2=(2m^2+2m+1)^2</math>

where integer <math>m>0</math> is the generating parameter.

It shows that all [[odd numbers]] (greater than 1) appear in this type of almost-isosceles primitive Pythagorean triple. This sequence of primitive Pythagorean triples forms the right hand side outer stem of the rooted ternary tree of primitive Pythagorean triples.

Another property of this type of almost-isosceles primitive Pythagorean triple is that the sides are related such that 
:<math>a^b+b^a=Kc</math>
for some integer <math>K</math>. Or in other words <math>a^b+b^a</math> is divisible by <math>c</math> such as in
:<math>(5^{12}+12^5)/13 = 18799189</math>.<ref>{{cite OEIS|A303734|mode=cs2}}</ref>