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September 1st, 2010, 01:30 PM   #1
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Prime and Fibonacci

Let P(n) be prime (P(1) = 2, P(2) = 3, P(3) = 5, P(4) = 7, P(5) = 11, . . .) and F(n) be Fibonacci (F(1) = 1, F(2) = 1, F(3) = 2, F(4) = 3, F(5) = 5, . . .). True or false:
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September 1st, 2010, 01:59 PM   #2
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Re: Prime and Fibonacci

P(n)<n

and

F(n) approx a^n/sqrt(5) (Binetīs form) with a=(1+sqrt(5))/2
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September 1st, 2010, 02:04 PM   #3
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Re: Prime and Fibonacci

Well, P(n) < n log n + n log log n (for n >= 6), but it works out the same.
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September 1st, 2010, 02:11 PM   #4
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Re: Prime and Fibonacci

Thatīs right:

Pi(n)<n but obviously P(n) not
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September 1st, 2010, 02:19 PM   #5
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Re: Prime and Fibonacci

The basic idea is the same, though: P(n) has essentially linear growth, while F(n) has exponential growth.
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September 1st, 2010, 05:11 PM   #6
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Quote:
Originally Posted by CRGreathouse
. . . P(n) < n log n + n log log n (for n >= 6) . . .
? As n ? ?, P(n)/F(n) = 0.
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