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A Prime Example of a Perfect Assignment Prime numbers were studied in ancient times for philosophical reasons. Today, there are now highly practical reasons for their study. In particular, large primes play a crucial role in cryptography. It has long been known that there are infinitely many primes (recall the indirect proof we went through). There is an ongoing quest to discover larger and larger prime numbers; for almost all the last 300 years, the largest prime known has been of the form 2^p - 1, where p is also prime. Such primes are called Mersenne Primes, after the French monk Marin Mersenne, who studied then in the seventeenth century. The reason the largest known prime has usually been a Mersenne prime is that there is an extremely efficient test, known as the Lucas-Lehmer test, for determining whether 2^p - 1 is prime. 1. Prove or disprove the following conjecture: 2^p - 1 is prime whenever p is prime. 2. Prove that if 2^n - 1 is prime, then n is prime. [Hint: First show that 2^ab - 1 = (2^a - 1)(2^a(b-1) + 2^a(b-2) + ... + 2^a + 1).] 3. A positive integer n is said to be perfect if n is equal to the sum of all its positive divisors, excluding n itself. For example, 6 is a perfect number since 6 = 1 + 2 + 3, as well as 28, since 28 = 1 + 2 + 4 + 7 + 14. The first few perfect numbers are: 6, 28, 496, 8128, 33550336, 8589869056 -- you are welcome to verify this list! There is an interesting relationship between Mersenne primes and perfect numbers. Whenever a new Mersenne prime is found, it yields a new perfect number! Prove that if 2^p - 1 is prime, then 2^(p-1)(2^p - 1) is a perfect number.

          A Prime Example of a Perfect Assignment

Prime numbers were studied in ancient times for philosophical reasons. Today, there are now highly practical reasons for their study. In particular, large primes play a crucial role in cryptography. It has long been known that there are infinitely many primes (recall the indirect proof we went through). There is an ongoing quest to discover larger and larger prime numbers; for almost all the last 300 years, the largest prime known has been of the form 2^p - 1, where p is also prime. Such primes are called Mersenne Primes, after the French monk Marin Mersenne, who studied then in the seventeenth century. The reason the largest known prime has usually been a Mersenne prime is that there is an extremely efficient test, known as the Lucas-Lehmer test, for determining whether 2^p - 1 is prime.

1. Prove or disprove the following conjecture: 2^p - 1 is prime whenever p is prime.

2. Prove that if 2^n - 1 is prime, then n is prime.
[Hint: First show that 2^ab - 1 = (2^a - 1)(2^a(b-1) + 2^a(b-2) + ... + 2^a + 1).]

3. A positive integer n is said to be perfect if n is equal to the sum of all its positive divisors, excluding n itself. For example, 6 is a perfect number since 6 = 1 + 2 + 3, as well as 28, since 28 = 1 + 2 + 4 + 7 + 14. The first few perfect numbers are: 6, 28, 496, 8128, 33550336, 8589869056 -- you are welcome to verify this list! There is an interesting relationship between Mersenne primes and perfect numbers. Whenever a new Mersenne prime is found, it yields a new perfect number! Prove that if 2^p - 1 is prime, then 2^(p-1)(2^p - 1) is a perfect number.

A Prime Example of a Perfect Assignment

Prime numbers were studied in ancient times for philosophical reasons. Today, there are now highly practical reasons for their study. In particular, large primes play a crucial role in cryptography. It has long been known that there are infinitely many primes (recall the indirect proof we went through). There is an ongoing quest to discover larger and larger prime numbers; for almost all the last 300 years, the largest prime known has been of the form 2^p - 1, where p is also prime. Such primes are called Mersenne Primes, after the French monk Marin Mersenne, who studied then in the seventeenth century. The reason the largest known prime has usually been a Mersenne prime is that there is an extremely efficient test, known as the Lucas-Lehmer test, for determining whether 2^p - 1 is prime.

1. Prove or disprove the following conjecture: 2^p - 1 is prime whenever p is prime.

2. Prove that if 2^n - 1 is prime, then n is prime.
[Hint: First show that 2^ab - 1 = (2^a - 1)(2^a(b-1) + 2^a(b-2) + ... + 2^a + 1).]

3. A positive integer n is said to be perfect if n is equal to the sum of all its positive divisors, excluding n itself. For example, 6 is a perfect number since 6 = 1 + 2 + 3, as well as 28, since 28 = 1 + 2 + 4 + 7 + 14. The first few perfect numbers are: 6, 28, 496, 8128, 33550336, 8589869056 – you are welcome to verify this list! There is an interesting relationship between Mersenne primes and perfect numbers. Whenever a new Mersenne prime is found, it yields a new perfect number! Prove that if 2^p - 1 is prime, then 2^(p-1)(2^p - 1) is a perfect number.

Added by Alfredo J.

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