Which of the following expressions could, by definition, yield a Mersenne prime number? Enter the number of the correct option.
A Mersenne prime number is a prime number that can be written in the form 2^p - 1, where p is a prime number.
Based on this definition, the only expression that could yield a Mersenne prime number is:
2^11 - 1
So the correct option is: 1
To identify which of the given expressions could yield a Mersenne prime number, let's first understand what a Mersenne prime is.
A Mersenne prime is a prime number that can be written in the form 2^n - 1, where n is a positive integer.
Now let's evaluate each expression:
1. 2^3 - 1 = 8 - 1 = 7
2. 2^5 - 1 = 32 - 1 = 31
3. 2^6 - 2 = 64 - 2 = 62
4. 2^9 - 1 = 512 - 1 = 511
So, the only expression that yields a Mersenne prime number is option 2, where 2^5 - 1 = 31.
Therefore, the correct option is 2.
In order to determine which of the given expressions could yield a Mersenne prime number, we need to 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.
Let's analyze each given expression to see if it fits the criteria for a Mersenne prime:
1. 2^3 - 1: This expression evaluates to 7. Since 7 is a prime number, it can be considered a Mersenne prime.
2. 2^4 - 1: This expression evaluates to 15. However, 15 is not a prime number, so it cannot be considered a Mersenne prime.
3. 2^5 + 1: This expression evaluates to 33. Again, 33 is not a prime number, so it cannot be considered a Mersenne prime.
4. 2^6 + 1: This expression evaluates to 65. Like the previous expressions, 65 is not a prime number, so it cannot be considered a Mersenne prime.
From the analysis above, we can see that only option 1, 2^3 - 1, yields a Mersenne prime number. Therefore, the correct option is 1.