
1,000 Years of Congruent Numbers - headalgorithm
https://blogs.scientificamerican.com/roots-of-unity/1000-years-of-congruent-numbers/
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dan-robertson
I found this article a little annoying to follow. Congruence is already a bit
of an overloaded term in mathematics and this didn’t help. I’ve worked out
some of the maths here.

An integer n is _congruent_ if it is the area of a right triangle with
rational sides, ie n = ab/2 where a^2 + b^2 = c^2 and a,b,c are positive
rational numbers.

The article then talks about another meaning of congruum: an arithmetic
progression of square natural numbers, x^2,y^2,z^2. In such a case y is the
hypotenuse of a right rational triangle. This is easy to prove:

    
    
      x^2 = y^2 - k
      z^2 = y^2 + k
      =>
      ((x+z)/2)^2 + ((z-x)/2)^2
      = x^2/2 + z^2/2
      = y^2
    

And the area of this triangle is:

    
    
      (x+z)(x-z)/8 = (x^2-z^2)/8 = k/4
    

\- - - -

There’s a lot of order and theorems about congruent numbers but it is hard (or
rather it is unknown whether it is hard) to determine if a number is
congruent. I don’t have much of a reason for there being lots of order other
than that there is a lot of order in the integers and rational numbers. I
think a reason for this problem being hard is the important word rational
above. I think if it said integer then the problem is only how to do better
than the trivial linear time solution of trying all sufficiently small right
angle triangles. Rationals mean that one needs to try larger (integer)
triangles as well as smaller ones as one can always divide by a rational
squared.

