While messing around, I noticed that across some prime numbers contain only sequentially increasing digits, e.g. $23, 67, 89,23456789$.
If we adopt a convention of returning to $1$ after a $9$, we can find some other primes, e.g. $67891,4567891,56789123,1234567891,4567891234567$.
Is there a name for prime numbers of this kind? Is it known if there are an infinite number of primes having this form?
Gerry Myerson below pointed me to a related OEIS sequence. This is different from the sequence above in that it returns to 0 after a 9, as opposed to 1 as outlined above. In hindsight, this seems like a more natural definition than mine.
These have been referred to as 'consecutive digit primes' in the literature. There is some literature published on this notably J. S. Madachy, Consecutive-digit primes - again, J. Rec. Math., 5 (No. 4, 1972), 253-254. But this doesn't seem to have been digitised yet.
I also found this yahoo question. Over there a user finds a 658-digit consecutive digit prime. The user that found this conjectured that there were a finite number of such primes. There is also a related question from here on stackexchange, which asks if there are any such consecutive digit primes with a first digit of 1, which there are.