- #1
Dragonfall
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Homework Statement
Prove that if a prime [tex]p|2^{2^n}+1[/tex] then [tex]p=2^{n+1}k+1[/tex] for some k.
Don't know how. I'm guessing by induction, perhaps?
The formula is 2^(2^n)+1 = (2^2)^n + 1.
2^(2^n)+1 is known as a Fermat number, named after the mathematician Pierre de Fermat. It is significant because it is closely related to Fermat's Last Theorem, one of the most famous unsolved problems in mathematics.
We can prove this using mathematical induction. First, we can show that 2^(2^1)+1 = 5 is divisible by 3. Then, assuming that 2^(2^n)+1 is divisible by 3, we can show that 2^(2^(n+1))+1 is also divisible by 3. Therefore, by induction, we can conclude that 2^(2^n)+1 is always divisible by 3.
Mersenne primes are prime numbers in the form of 2^n-1, where n is a positive integer. Interestingly, 2^(2^n)+1 is a Mersenne number when n is equal to 1. However, for n>1, 2^(2^n)+1 is not a Mersenne number and may or may not be prime.
Proving the divisibility of 2^(2^n)+1 is important because it helps us understand the properties of prime numbers and their relationships with other numbers. It also has practical applications in cryptography and number theory.