Does there exist a prime that is only consecutive digits starting from 1?
Edit: code fixed, uses gmpy2 now.
The power of brute force: I wrote a quick python program, and
123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901
is a prime based on gmpy2's probable prime Miller-Rabin test.
Code if you want to verify:
import gmpy2
digit = 1
number = 1234567890
while True:
number = 10*number + digit
if gmpy2.is_prime(number):
print(number)
break
digit = (digit + 1)%10