Understanding Perfect Numbers and Mersenne Primes

The video explores the concept of perfect numbers, which are numbers equal to the sum of their proper factors, and introduces Mersenne primes, a special class of prime numbers. It discusses the patterns observed in Mersenne primes and their connection to perfect numbers, highlighting that every even perfect number has a Mersenne prime factor. The video also delves into logical reasoning, explaining that while not all numbers of the form 2^n - 1 are prime, if such a number is prime, then n must be prime. The video concludes by emphasizing the link between Mersenne primes and perfect numbers, which is crucial for discovering new perfect numbers.

What is the smallest perfect number?

• 10

• 3

• 15

• 6

Which of the following is a perfect number?

• 50

• 12

• 28

• 45

What is a Mersenne prime?

• A prime number that is a sum of two powers of two

• A prime number that is a power of two

• A prime number that is one more than a power of two

• A prime number that is one less than a power of two

Which value of n gives a Mersenne prime when using the formula 2^n - 1?

• n = 4

• n = 7

• n = 5

• n = 6

What is the relationship between Mersenne primes and perfect numbers?

• Mersenne primes and perfect numbers are unrelated

• Perfect numbers are always greater than Mersenne primes

• Every Mersenne prime has a corresponding perfect number

• Every perfect number is a Mersenne prime

Which of the following is true about the pattern of Mersenne primes?

• If 2^n - 1 is prime, then n must be prime

• If n is prime, then 2^n - 1 is always prime

• 2^n - 1 is never prime

• 2^n - 1 is always prime

What is the next Mersenne prime after 31?

• 511

• 255

• 63

• 127

How are new perfect numbers typically discovered?

• By finding new Mersenne primes

• By calculating large factorials

• By summing consecutive integers

• By dividing large numbers by small primes

What was the most recent year a new perfect number was announced?

• 2010

• 2015

• 2013

• 2018

What is the significance of the number 8128 in the context of perfect numbers?

• It is a Mersenne prime

• It is a perfect number with a Mersenne prime factor

• It is the largest known perfect number

• It is the smallest perfect number