a2+b2=c2 and ab/2=2008


     Cool!!! 2008 is a congruent number!


     
     Substitute n=2008 to y2=x3-n2x and solve about (x,y),then we obtain
     (x,y)=(-33189589138608236/173526479973025,-63190422991165235540565912/2285854776728259808625)

     a=(n2-x2)/y=10946706347066226/75738545443585

     b=2nx/y=605908363548680/21806187942363

     c=(n2+x2)/y=243077092075274283595010477962/1651568956424015360248091355

     a2+b2=c2 and ab/2=2008



     When I used the Tunnell's Theorem,I noticed a certain pattern.

     A certain pattern is ,if n = 5,6,7 mod 8, then n becomes a congruent number.

     I noticed that the corresponding equations always have no solutions.

     Therefore, such number n = 5,6,7 mod 8 is always a congruent number by Tunnell's Theorem. 

     I proved this pattern which I noticed as a next lemma.

     Consequently,if we check only whether n is congruent number or not by Tunnell's theorem,
     we can exclude any natural number n=5,6,7 mod 8 from cheking.


     Tunnell's Theorem

         Let n be a squarefree natural number.

         If n is odd then n is congruent number,then
         the number of solutions of x2+2y2+8z2 =n is equal to the number of solutions of
         x2+2y2+32z2=n.

         If n is even then n is congruent number,then
         the number of solutions of x2+4y2+8z2 =n/2 is equal to the number of solutions of
         x2+4y2+32z2=n/2.

         Furthermore,if the Birch and Swinnerton-Dyer conjecture holds,then
         the converse also holds.

      Lemma

        Any natural number n=5,6,7 mod 8 is congruent number
       if the Birch and Swinnerton-Dyer conjecture holds.
       

        Proof.
    
        (1). Squarefree natural number n=5,7 mod 8 is congruent number.

             if n is squarefree natural number and odd,examine that 
             the number of solutions of x2+2y2+8z2 =n and number of solutions of
             x2+2y2+32z2=n.

             x2={0,1,4} mod 8,so n=x2+2y2={0,1,2,3,4,6} mod 8
             So,if n=5,7 mod 8 then x2+2y2+8z2=n and x2+2y2+32z2=n has no solution.
             Therefore,the number of solutions of x2+2y2+8z2 =n is equal to the number of
             solutions of x2+2y2+32z2=n.
             By Tunnell' theorem,squarefree natural number n=5,7 mod 8 is congruent number.


        (2). Squarefree natural number n=6 mod 8 is congruent number.

             if n is squarefree natural number and even,examine that 
             the number of solutions of x2+4y2+8z2 =n/2 and number of solutions of
             x2+4y2+32z2=n/2.
             In the same way as the odd case,n/2=x2+4y2={0,1,4,5} mod 8
             So,if n=6 mod 8 x2+4y2+8z2 =n/2 and x2+4y2+32z2=n/2 has no solution.
             Therefore,the number of solutions of x2+4y2+8z2 =n/2 is equal to the number of
             solutions of x2+4y2+32z2=n/2.
             By Tunnell' theorem,squarefree natural number n=6 mod 8 is congruent number.

        (3). If natural number d2n=5,6,7 mod 8 then n must be 5,6,7 mod 8.

             if d2n=5 mod 8 then d =1 mod 8,so n=5 mod 8.

             In the same way ,n=5,6,7 mod 8.


        Consequently, by (1),(2),and (3), Any natural number n=5,6,7 mod 8 is congruent number.