How many pairs of twin primes are there between the integers 1 to 100?

Twin primes are pairs of prime numbers that differ by exactly two, such as (3, 5) or (11, 13). To count how many such pairs exist in a range, every prime number in that range is listed and each consecutive pair of primes is checked for a gap of exactly two — a prime can belong to two different twin-prime pairs when it sits between two other primes each two apart from it, and every qualifying gap is counted as its own pair.

Listing the primes from 2 through 97 gives: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. Checking each consecutive pair for a difference of two identifies these twin-prime pairs:

1. (3, 5)

2. (5, 7)

3. (11, 13)

4. (17, 19)

5. (29, 31)

6. (41, 43)

7. (59, 61)

8. (71, 73)

Every other consecutive gap among these primes is not equal to two — for instance, 2 to 3 differs by one, 7 to 11 differs by four, 23 to 29 differs by six, and 89 to 97 differs by eight — confirming that no further pairs qualify beyond the eight identified above.