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On congruent numbers (Posted on 2013-02-24) Difficulty: 4 of 5
An integer n is a congruent number iff there exists a right triangle having area n and rational sides. (Sloane A003273)
Show that the numbers 5,6,7 are congruent and 8 is not.

Determine how to decide whether a given number is congruent or not.

No Solution Yet Submitted by Ady TZIDON    
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Some Thoughts Possible Approach Comment 2 of 2 |

To cut a long story short: 6 is the simply the area of the 3,4,5 triangle. Solutions for 5 and 7 were provided in Conga Primes (there is a close connection between congruum numbers and congruent numbers, see also http://perplexus.info/show.php?pid=8275).

As to 8:

(1)  The 'congruum part', d, always conforms to the equality 4xy(x-y)(x+y)=d, and a triple will be available iff xy^3-x^3y = (6a)^2(K),  with K the congruent number (which in this case need not be prime).

(2)  If K contains a square, the square can be absorbed into the squared part of this equation, giving a smaller solution for some 'primitive'  squarefree K. 

(3)  If we write xy^3-x^3y as xy(y^2-x^2), we immediately have the classic formula for the area of a right triangle with sides (y^2-x^2) and (2xy). But this is the same as Fermat's 'triangle of numbers' problem - with no solutions for xy(y^2-x^2) a square or twice a square.

(4)  Now assume that 8 was congruent. We could absorb the square part, 4, leaving a 'primitive' solution for K=2; but that is not possible. So 8 is not congruent.

(5)  The last part of the problem is not straightforward.  One source states: 'Along with the Riemann Hypothesis the Birch and Swinnerton-Dyer Conjecture is one of the seven Millenium Prize Problems posed by the Clay Institute with a prize of one million dollars for the solution of each. For each positive square-free number n another number is computed by Tunnell's formula. If this number is not zero then n is not a congruent number, but if this computed number is zero then n is congruent as long as the BSD Conjecture is true. Tunnell's formula can even be computed by hand for small values of n, and it can be computed easily for larger values using a personal computer, but as the numbers get even larger the computations cannot be done with ordinary computers and the typical mathematical software available.'

Nice try, Ady.
 

 

Edited on April 18, 2013, 1:37 am
  Posted by broll on 2013-04-18 01:32:04

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