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
+118608 I'd be interested in seeing an answer (or an outline) to this one too. 
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.
