Fermat Numbers

The Fermat Numbers form a sequence in the form

Fn = (2^(2^n)) + 1, n = 0, 1, 2, …
or Fn = pow(2,(pow(2,n))) + 1

The number of digits for a Fermat number is
D(Fn) = └[log (2^(2^n))+ 1)] + 1┘
D(Fn) = └[log (2^(2^n))] + 1┘
D(Fn) = 1 + └2^n log 2┘
or D(Fn) = 1 + floor((2^n) log 2)

The first few Fermat numbers are:

F0 = 2^(2^0) + 1 = (2^1) + 1 = 2 + 1 = 3
F1 = 2^(2^1) + 1 = (2^2) + 1 = 4 + 1 = 5,
F2 = 2^(2^2) + 1 = (2^4) + 1 = 16 + 1 = 17,
F3 = 2^(2^3) + 1 = (2^8) + 1 = 256 + 1 = 257,
F4 = 2^(2^4) + 1 = (2^16) + 1 = 65536 + 1 = 65537

The corresponding number of digits:

D(F0) = 1 +└(2^0) log 2┘= 1 +└1 log 2┘= 1 + 0 = 1
D(F1) = 1 +└(2^1) log 2┘= 1 +└2 log 2┘= 1 + 0 = 1
D(F2) = 1 +└(2^2) log 2┘= 1 +└4 log 2┘= 1 + 1 = 2
D(F3) = 1 +└(2^3) log 2┘= 1 +└8 log 2┘= 1 + 2 = 3
D(F4) = 1 +└(2^4) log 2┘= 1+└16 log 2┘= 1+ 4 = 5

These numbers are more popularly known as Fermat primes.
F5, F6, F7, F8, F9, F10 and F11 are factorable numbers. F11 being the largest known Fermat number to be factored out.

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