Pensieve Header: Brute force "The Strong Law of Small Numbers" (http://www.math.sjsu.edu/~hsu/courses/126/Law-of-Small-Numbers.pdf), for no reason at all.

Example 1

Do[
Print[
k-> FactorInteger[2^(2^k)+1]
],
{k,0,5}
]

Example 2

Do[
Print[
Prime[k]-> FactorInteger[2^Prime[k]-1]
],
{k,5}
]

Example 3

Stats[n_] := Length[Select[Prime /@ Range[n], (1==(#~Mod~4))&]];
Stats[100000]

Draw[n_] := Rasterize[ListPlot[
t=0;
Table[
t += If[(Prime[k]~Mod~4) == 1, 1, -1],
{k, n}
]
]];
Draw[100000]

Rasterize[ListPlot[
t=0;
Take[Table[
t += If[(Prime[k]~Mod~4) == 1, 1, -1],
{k, 100000}
], {50000, 52000}]
]]

2Stats[50881]

Example 4

Plus @@ Table[
divs = Divisors[k];
((l=Length[Select[divs, 3==#~Mod~4 &]]) ≤ Length[divs]-l),
{k, 1, 100001, 2}
]

Example 5