3 ms·
1,000 Years of Congruent Numbers
- dan-robertson 7y agoI 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.