The Twin Prime Conjecture, which
has not been proven, argues that
there are infinitely many pairs of
primes of the form n and n+2. Such
a pair is called a prime pair. Some
examples of prime pairs are 5 and
7, 17 and 19, and 29 and 31. How
many prime pairs have both
members less than 100?
To find out how many prime pairs have both members less than 100, we can check each number from 1 to 100 and see if it is a prime number and if its adjacent number (n+2) is also a prime number. If both conditions are true, then we have a prime pair.
Here's how we can do this:
1. Start by initializing a counter variable to keep track of the number of prime pairs. Let's call it "count" and set it to 0.
2. Iterate through each number from 1 to 100.
3. For each number, check if it is prime. To determine if a number is prime, check if it is divisible by any number from 2 up to its square root. If it is divisible, then it is not prime. If it is not divisible by any number in that range, then it is prime.
4. Once you determine that a number is prime, check if its adjacent number (n+2) is also prime using the same prime-checking method as step 3.
5. If both the number and its adjacent number are prime, increment the counter variable "count" by 1.
6. Repeat steps 3 to 5 for all numbers in the range.
7. Finally, when you finish iterating through all numbers, the value of the counter variable "count" will be the number of prime pairs with both members less than 100.
Implementing this algorithm, we find that there are 8 prime pairs with both members less than 100: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), and (71, 73).
Please note that the Twin Prime Conjecture, which states that there are infinitely many prime pairs of the form n and n+2, has not been proven yet. However, it has been tested for larger numbers and many prime pairs have been found, indicating that it is likely true.