The difference between consecutive prime numbers is called a prime gap.
If pₙ and pₙ₊₁ are successive primes, their gap is written gₙ = pₙ₊₁ − pₙ. The first primes are 2, 3, 5, 7, and 11, producing gaps of 1, 2, 2, and 4. The word “consecutive” matters: the numbers must have no other prime between them.
The only odd prime gap is 1, between 2 and 3. Every later prime is odd, so the difference between two successive odd primes is even. Prime gaps can become arbitrarily large; factorial constructions show that long runs of composite numbers must occur somewhere.
The average gap near a large prime grows roughly like its natural logarithm, a consequence of the prime number theorem, but individual gaps vary widely. The twin prime conjecture asks whether a gap of 2 occurs infinitely often. Goldbach’s conjecture concerns sums of primes, while the prime number theorem describes their overall distribution; neither is another name for a prime gap.