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I was inspired by this question, in particular, where removing any digit of a prime yielded another prime. My question is, are there any numbers that this holds true for, and will continue holding true for if digits continue to be removed?

I have a strong feeling that this is false. However, I'm very interested in seeing a proof for the falsehood of this, or maybe even a proof that one does exist but is not computable within reason.

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If you're asking what I think you're asking, it shouldn't be hard to work through it. First, since you eventually get down to one digit, only prime numbers can be digits; 2, 3, 5, and 7. But 2 and 5 are no good, since any 2-digit number ending in 2 or 5 will be composite, so we just have 3 and 7. But we can't use either digit twice, since if we remove all the other digits we'll get 33 or 77, both composite. So Matthew's 37 (and 73) is the only answer of more than one digit.

If you choose to make believe that 1 is a prime, then you also get 13 (and 31) and 17 (and 71), and you have to check whether the numbers 137, 173, 317, 371, 713, and 731 are all prime.

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    $\begingroup$ You didn't take into account that 2 or 5 could occur at the front, where they would never be at the end of a 2-digit number. That leads to the additional possibilities 23, 27, 237, 273, 53, 57, 537, 573, of which 23 and 53 are of the desired form and the rest are multiples of 3. $\endgroup$ – joriki May 23 '11 at 11:46
  • $\begingroup$ @joriki, you and I are taking different readings of the unclear statement of the problem. "any in-order combination of digits is prime (including the original number)" I took to mean re-arranging 23 to 32 would be included so 23 wouldn't work. But maybe that's not what Eugene meant. $\endgroup$ – Gerry Myerson May 23 '11 at 13:36
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    $\begingroup$ It seems clear to me (especially if you look at the original question it's derived from) that you can delete any digits you like, but you can't rearrange them. That's the "in-order" part. So 23 and 53 are also valid answers. $\endgroup$ – cjm Mar 15 '14 at 5:43
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How about 37? 3 is prime, 7 is prime. 73 is prime, too, if you want to reorder the digits.

Also, see truncatable primes..

Added: 23 works, too, if you don't want to reorder things.

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