Question

Use exercise 41 to determine whether m₃₁₃ = 2¹³ - 1 = 8191 and m₂₃ = 2²³ - 1 = 8,388,607 are prime.

a) m₃₁₃ is prime, m₂₃ is composite

b) m₃₁₃ is composite, m₂₃ is prime

c) Both m₃₁₃ and m₂₃ are prime

d) Both m₃₁₃ and m₂₃ are composite

Answer 1

The number m₃₁₃ or 8191 is a proven prime, known as a Mersenne prime, while m₂₃ or 8,388,607 is a composite number with factors 47 and 178,481.

The correct answer is (a) m₃₁₃ is prime, m₂₃ is composite.

Step-by-step explanation:

The student's question involves determining whether m₃₁₃ = 2¹³ - 1 = 8191 and m₂₃ = 2²³ - 1 = 8,388,607 are prime numbers. In the context of number theory, numbers of the form Mn = 2ⁿ - 1 are known as Mersenne primes.

For m₃₁₃, it is known to be a prime number. Therefore, the number 8191 is prime, which corresponds to part (a) or (c) of the potential answers provided by the student.

Regarding m₂₃, it is not a prime number. Its factors include 47 and 178,481, making the number 8,388,607 a composite number. This confirms that option (b) is incorrect, and we are left with option (a) as m₃₁₃ is prime and m₂₃ is composite.

The correct answer to the student's question is:

• (a) m₃₁₃ is prime, m₂₃ is composite