I recently saw a video about the last digit of prime numbers, that if a prime ends with a digit X then it is the least likely that the next prime also has X as the last digit.
But I counted the prime gaps (modulo 10) for primes between 200 million and 2 billion and the result is this:
- 0: 15885351
- 2: 19886835
- 4: 17246764
- 6: 18937346
- 8: 15265989
Why 0 is not the least common prime gap modulo 10? I expected it to be significantly less then any other gaps.
Edit1: Link to the numberphile video
Edit2: below the counts how many times the digit on the left is followed by the digit on the top (for the last digit of primes between 200 million and 2 billion):
1 3 5 7 9
1 4047130 6467273 0 6526597 4763818
3 5250387 3896688 0 6133642 6525043
5 0 0 0 0 0
7 5557028 5885706 0 3893204 6469289
9 6950273 5556094 0 5251784 4048329
You see the counts in the main diagonal are much lower than any other.
There are only three last digits that can lead to a gap of $8$ while there are four that can lead to a gap of $10$. If the last digit of a prime is $7$, a gap of $8$ would lead to a number whose last digit is $5$, which cannot be prime. This suppresses gaps of $8$.