# deletable prime

If we can delete the digits from N one at a time from the right and still get a prime, then N is a right truncatable prime.  If we can delete the digits from N one at a time from the left and still get a prime, then N is a left truncatable prime.  Are there any primes in which we can repeatedly delete any digit and still get a prime at each step? If so, each digit would have to be prime, and no digit could occur twice, so this would be a short list: 2, 3, 5, 7, 23, 37, 53 and 73.

To make the search more interesting, a deletable prime has been defined ([Caldwell87]) to be a prime that you can delete the digits one at a time in some order and get a prime at each step.  One example is 410256793, because the following are (deletable) primes:

• 410256793
• 41256793
• 4125673
• 415673
• 45673
• 4567
• 467
• 67
• 7
It is conjectured that there are infinitely many of these primes (and this may be one of the easiest conjectures in this glossary to prove!)

References:

Caldwell87
C. Caldwell, "Truncatable primes," J. Recreational Math., 19:1 (1987) 30--33. [A recreational note discussing left truncatable primes, right truncatable primes, and deletable primes.]