Which of the following statements must be true about n in the expression for a Mersenne prime, 2n−1

Question

Which of the following statements must be true about n in the expression for a Mersenne prime, 2n−1

?

Answer 1

The expression for a Mersenne prime is 2^𝑛−1. Therefore, the value of n must be a positive integer.

Answer 2

Question

Mei wants to generate a Pythagorean triple when x equals 8 and y equals 6 using the polynomial identity (x^2+y^2)^2=(x^2−y^2)^2+(2xy)^2
. How does she write the Pythagorean triple, in ascending order?

Answer 3

To generate a Pythagorean triple using the polynomial identity (x^2+y^2)^2=(x^2−y^2)^2+(2xy)^2, we can substitute the given values for x and y.

When x = 8 and y = 6, the equation becomes: (8^2 + 6^2)^2 = (8^2 - 6^2)^2 + (2*8*6)^2

Simplifying this equation gives: (64 + 36)^2 = (64 - 36)^2 + (96)^2

This further simplifies to: 100^2 = 28^2 + 96^2

Therefore, the Pythagorean triple can be written as (28, 96, 100) in ascending order.

Answer 4

To determine which statements must be true about n in the expression for a Mersenne prime, 2n-1, we need to consider the properties of Mersenne primes.

1. n must be a positive integer: This is true because n represents the exponent in the expression 2^n.

2. n must be prime: This statement must be true. According to the definition of Mersenne primes, n must be a prime number in order for 2^n - 1 to be a prime number.

3. n must be an odd number: This statement must be true. All Mersenne primes are of the form 2^n - 1, and since the existence of an even Mersenne prime has not been proven, it is generally understood that n must be an odd number.

Therefore, the statements that must be true about n in the expression for a Mersenne prime, 2n-1, are:
- n must be a positive integer
- n must be prime
- n must be an odd number.

Answer 5

To determine which statements about n in the expression for a Mersenne prime, 2n−1, must be true, let's first understand what a Mersenne prime is.

A Mersenne prime is a prime number that can be expressed in the form 2^n - 1, where n is a positive integer. In other words, it is a prime number that is one less than a power of 2.

Now, let's consider the statements:

1. n is odd.
2. n is even.
3. n is a prime number.
4. n is a composite number.
5. n is greater than 1.

To determine which statements must be true, we can analyze each one:

1. n is odd: This statement must be true because if n were even, then 2^n - 1 would also be even. However, by definition, Mersenne primes are odd.

2. n is even: This statement cannot be true since Mersenne primes are defined as 2^n - 1, where n is a positive integer. If n were even, then 2^n - 1 would be odd, not a Mersenne prime.

3. n is a prime number: This statement is not necessarily true. While it is true for some Mersenne primes, not all values of n that produce a Mersenne prime are prime numbers.

4. n is a composite number: This statement is not necessarily true. Similarly to the previous statement, some Mersenne primes can be generated by composite numbers.

5. n is greater than 1: This statement must be true since Mersenne primes are defined using the expression 2^n - 1, where n is a positive integer. If n were 0 or negative, the expression would not produce a prime number.

In conclusion, the statements that must be true about n in the expression for a Mersenne prime, 2^n - 1, are:
- n is odd.
- n is greater than 1.