Perhaps surprisingly, given any sequence of digits whatsoever, there are infinitely many prime numbers that start off with that sequence. For example, take 2026 as the sequence. Here are 19 primes beginning 2026
20261, 202613, 2026021, 20260001, 202600043, 2026000003, 20260000009, 202600000033, 2026000000021, 20260000000013, 202600000000009, 2026000000000007, 20260000000000021, 202600000000000003, 2026000000000000001, 20260000000000000033, 202600000000000000009, 2026000000000000000033, 20260000000000000000003.
There are lots of others but I've chosen ones that grow by a order of magnitude each time because I think it suggests an argument that we can go on finding similar primes for ever.
Consider the
9 integers 20261 - 20269
99 integers 202601 - 202699
999 integers 2026001 - 2026999
and so on. Now a good estimate of the average gaps between consecutive primes near a given is known to be (the natural logarithm of . If we divide the sizes of the intervals, 9, 99, 999... by the logarithms of 20265, 202650, 2026500... (the midpoints of the ranges) that should give the approximate number of primes we can expect to find in each range. This gives
20261 - 20269 estimate 1 actual 1
202601 - 202699 estimate 8 actual 8
2026001 - 2026999 estimate 69 actual 72
The estimates are pretty good! The numbers are actually growing, which more than supports the contention that primes starting with 2026 are infinite in number. This can actually be proved properly; my heuristic above seems plausible but is not an actual proof.