Mersenne prime is a number of the form Mn = 2^n − 1 for some prime n. Fermat number is defined as F[n] = 2^n +1 where ‘n’ is a natural number greater than zero, this form is (less commonly) used. The more popular Fermat numbers are defined as F[n] = 2^2^n + 1 where ‘n’ is a natural number greater than or equal to zero. Relations between these types of primes, if any, is useful in research related to prime numbers.
See the attachment for a relation between Mersenne primes and Fermat primes.
Dear Adiya K. Hussein ,
The Lucas - Lehmer ( with many modified versions) is one of the best to test if a Mersenne number 2 n - 1 is prime.
A similar version to test the primality of Fn = 2^(2 n ) + 1 :
Let S(i) = Si-14 - S2i-1 + 2 , S(0) = 8.
The author (2012) proved that
Fn is prime if and only if S(2 n-1 - 1) ≡ 0 mod (Fn).
See: vixra.org/pdf/1202.0029v1.pdf
Best regards
If you consider an integer as made up of two parts 1. The right end digit and 2. The remaining digits, n then 321 has n=32 and R = 1. Since n= 3t+2 and R= 1 321 can nevver be prime. My colleague prof A.G. Shannon and I have developed this theory to show primes not random. See our book. Integer Structure and Fibonacci Numbers from Nova Scientific Publishers. New York 1918.
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