Which of the following numbers is a prime in the form 2^n−1?
A prime number in the form $2^n-1$ is called a Mersenne prime. We need to determine which of the given numbers can be expressed in the form $2^n - 1$ for some positive integer $n$. Checking each of the numbers $287$, $431$, and $577$, we find that $431$ can be expressed as $2^8 - 1 = 255$, which is a Mersenne prime. Therefore, $\boxed{431}$ is a prime in the form $2^n - 1$.
Which of the following numbers is a prime in the form 2n−1?
The prime numbers in the form $2n-1$ are called Sophie Germain primes. We need to determine which of the given numbers can be expressed in the form $2n-1$ for some positive integer $n$. Checking each of the numbers $287$, $431$, and $577$, we find that none of them can be expressed in this form. Therefore, there is no prime in the form $2n-1$ among the given numbers.
To find a prime number in the form 2^n - 1, we need to check if the value obtained for different values of 'n' is prime. Here are the steps to identify which of the given numbers in the form 2^n - 1 are prime:
1. Check the values of 'n' for each number given and determine the corresponding expression in the form 2^n - 1.
2. For each expression, substitute 'n' and calculate the value.
3. Check if the obtained value is a prime number.
Here are the calculations for the given numbers:
a) 2^2 - 1 = 4 - 1 = 3
b) 2^3 - 1 = 8 - 1 = 7
c) 2^4 - 1 = 16 - 1 = 15
d) 2^5 - 1 = 32 - 1 = 31
Now, we need to determine which of these numbers are prime.
- 3 is a prime number
- 7 is a prime number
- 15 is not a prime number; it is divisible by 3 and 5
- 31 is a prime number
Therefore, the prime numbers in the form 2^n - 1 are 3, 7, and 31.
To find out which of the given numbers in the form 2^n - 1 are prime, we need to check each number and apply a test called the "Lucas-Lehmer test."
The Lucas-Lehmer test is a primality test specifically designed for numbers of the form 2^n - 1. It works efficiently for these numbers, making it an ideal choice for our current situation.
Here's a step-by-step process to apply the Lucas-Lehmer test for each given number:
1. Start with the number n that is given in the form 2^n - 1.
2. If n = 2, the corresponding number is 3, which is a prime number. So, if the given number is 3, it is prime in the form 2^n - 1.
3. For larger values of n, apply the Lucas-Lehmer test:
a. Calculate the value of s0 as follows:
s0 = 4
b. Iterate through a loop n - 2 times, updating the value of s on each iteration:
for i = 0 to n - 2:
si = (si-1)^2 - 2
c. After completing the loop, check if sn - 2 is divisible by the given number in the form 2^n - 1.
4. If sn - 2 is divisible by the given number, then the number is composite.
5. If sn - 2 is not divisible by the given number, then the number is prime in the form 2^n - 1.
By applying the Lucas-Lehmer test to each of the given numbers, you can determine which ones are prime in the form 2^n - 1.