Prove that there exists a positive integer N such that there are at least 2005 ordered pairs (x,y), of non-negative integers x and y satisfying x^2 + y^2 = N


[Help would be much appreciated! Thank you everyone!](/questions/help-would-be-much-appreciated-thank-you-everyone)
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I'd be interested in seeing an answer (or an outline) to this one too. ![indecision](/img/emoticons/smiley-undecided.gif "indecision")


I'm reasonably certain that 2005^2005 has AT LEAST 2005 pairs of(x, y) that will satisfy:

 x^2 + y^2 = N, but don't know how to prove it mathematically!!. The reason I say this is because I wrote a short computer code that shows 2005 has the following pairs:

  

 1  -  18 ^2  +  41 ^2 =  2005^1  
 2  -  22 ^2  +  39 ^2 =  2005  
 3  -  39 ^2  +  22 ^2 =  2005  
 4  -  41 ^2  +  18 ^2 =  2005

  

  

 1  -  200 ^2  +  1995 ^2 =  4020025=2005^2  
 2  -  1037 ^2  +  1716 ^2 =  4020025  
 3  -  1203 ^2  +  1604 ^2 =  4020025  
 4  -  1357 ^2  +  1476 ^2 =  4020025  
 5  -  1476 ^2  +  1357 ^2 =  4020025  
 6  -  1604 ^2  +  1203 ^2 =  4020025  
 7  -  1716 ^2  +  1037 ^2 =  4020025  
 8  -  1995 ^2  +  200 ^2 =  4020025

  

  

 1  -  2691 ^2  +  89738 ^2 =  8060150125 =2005^3  
 2  -  11629 ^2  +  89022 ^2 =  8060150125  
 3  -  20451 ^2  +  87418 ^2 =  8060150125  
 4  -  27710 ^2  +  85395 ^2 =  8060150125  
 5  -  29069 ^2  +  84942 ^2 =  8060150125  
 6  -  36090 ^2  +  82205 ^2 =  8060150125  
 7  -  44110 ^2  +  78195 ^2 =  8060150125  
 8  -  51690 ^2  +  73405 ^2 =  8060150125  
 9  -  73405 ^2  +  51690 ^2 =  8060150125  
 10  -  78195 ^2  +  44110 ^2 =  8060150125  
 11  -  82205 ^2  +  36090 ^2 =  8060150125  
 12  -  84942 ^2  +  29069 ^2 =  8060150125  
 13  -  85395 ^2  +  27710 ^2 =  8060150125  
 14  -  87418 ^2  +  20451 ^2 =  8060150125  
 15  -  89022 ^2  +  11629 ^2 =  8060150125  
 16  -  89738 ^2  +  2691 ^2 =  8060150125

  

  

  

  

  

 1  -  337127 ^2  +  4005864 ^2 =  1616 0601000625=2005^4  
 2  -  401000 ^2  +  3999975 ^2 =  1616 0601000625  
 3  -  735033 ^2  +  3952256 ^2 =  1616 0601000625  
 4  -  798000 ^2  +  3940025 ^2 =  1616 0601000625  
 5  -  1125607 ^2  +  3859224 ^2 =  1616 0601000625  
 6  -  1504953 ^2  +  3727696 ^2 =  1616 0601000625  
 7  -  1725615 ^2  +  3630820 ^2 =  1616 0601000625  
 8  -  1869287 ^2  +  3558984 ^2 =  1616 0601000625  
 9  -  2079185 ^2  +  3440580 ^2 =  1616 0601000625  
 10  -  2412015 ^2  +  3216020 ^2 =  1616 0601000625  
 11  -  2673220 ^2  +  3002415 ^2 =  1616 0601000625  
 12  -  2720785 ^2  +  2959380 ^2 =  1616 0601000625  
 13  -  2959380 ^2  +  2720785 ^2 =  1616 0601000625  
 14  -  3002415 ^2  +  2673220 ^2 =  1616 0601000625  
 15  -  3216020 ^2  +  2412015 ^2 =  1616 0601000625  
 16  -  3440580 ^2  +  2079185 ^2 =  1616 0601000625  
 17  -  3558984 ^2  +  1869287 ^2 =  1616 0601000625  
 18  -  3630820 ^2  +  1725615 ^2 =  1616 0601000625  
 19  -  3727696 ^2  +  1504953 ^2 =  1616 0601000625  
 20  -  3859224 ^2  +  1125607 ^2 =  1616 0601000625  
 21  -  3940025 ^2  +  798000 ^2 =  1616 0601000625  
 22  -  3952256 ^2  +  735033 ^2 =  1616 0601000625  
 23  -  3999975 ^2  +  401000 ^2 =  1616 0601000625  
 24  -  4005864 ^2  +  337127 ^2 =  1616 0601000625

  

 I believe that for every 1 increase in the power of 2005 will have a minimum increase of 4 pairs of x^2 + y^2, but as I said I don't know how to prove it mathematically! Maybe Alan, heureka or other mathematicians.

