The differences between consecutive prime numbers are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6, 12.
The only prime numbers that have an odd difference are the first two prime numbers two and three. Twin primes have a difference of two, cousin primes have a difference of four, and sexy primes have a difference of six.
Prime triplets have differences of either 2,4 or 4,2. Prime quadruplets have differences of 2,2,4 or 2,4,2 and prime quintuplets have differences of 2,4,2,4 or 4,2,4,2. Finally, prime sextuplets have differences 4,2,4,2,4. The twin primes conjecture states that the number of differences of 2 are infinite which is necessary for there to be an infinite number of these prime k-tuples.
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