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Mersenne Primes (3/24)

Mersenne Primes (3/24). Question: Under what conditions on a and n is it possible for a n – 1 to be prime? First, observe, using basic algebra, that a must be 2. Second, observe that even if a = 2, 2 n – 1 may or may not be prime. Look at some data.

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Mersenne Primes (3/24)

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  1. Mersenne Primes (3/24) • Question: Under what conditions on a and n is it possible for an – 1 to be prime? • First, observe, using basic algebra, that a must be 2. • Second, observe that even if a = 2, 2n– 1may or may not be prime. Look at some data. • Conclude, using algebra, that if n is composite, then2n– 1 is also composite. • So we have the following: If a 2 or n is composite, thenan– 1 is composite. • Carefully state to contra-positive of this. • Finally, is the converse of this statement true?? Check data.

  2. Definition and Questions • Definition. If p is prime, then 2p – 1is called a Mersenne number. If it is prime, then it is called a Mersenne prime. • It is important to remember that not all Mersenne numbers are Mersenne primes. The smallest example is 211 – 1 = 2047 = (23)(89). • Question: What is the largest known Mersenne prime as of today? • Google it! • Bigger Question: Are there infinitely many Mersenne primes? • Unknown! • For Wednesday: Read Chapter 14 and do all three exercises.

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