Did I do it?
Number Theory
Level
2
There exists a number, 102400...002401, where the 0's occur 2014 times. Is this number prime or composite?
Composite
Both prime and composite
Prime
Neither prime nor composite
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I'm taking this to mean that there are 2012 zeroes in the middle, between 1024 and 2401. In that case, our number is 1 0 2 4 ⋅ 1 0 2 0 1 6 + 2 4 0 1 = 4 ( 2 5 6 ⋅ 1 0 2 0 1 6 ) + 7 4 = 4 ( 4 ⋅ 1 0 5 0 4 ) 4 + 7 4 , and numbers of the form 4 x 4 + y 4 factor nontrivially by the Sophie Germain identity 4 x 4 + y 4 = ( 2 x 2 + 2 x y + y 2 ) ( 2 x 2 − 2 x y + y 2 ) .