Primes In the Decimal Expansion of p
Letting pk denote the integer given by the k most significant decimal digits of p, it's not too difficult to determine that pk is prime for k = 1, 2, 6, and 38. In other words, the following integers are primes
p
1 = 3p
2 = 31p
6 = 314159p
38 = 31415926535897932384626433832795028841Maple's probabilistic primality test shows that pk is composite for every other value of k from 1 to 500. A deterministic test of primality for numbers of more than 500 digits is fairly challenging, so it will be difficult to rigorously prove the primality of any further terms in this sequence.
It's interesting to consider the "expected number" of primes in this list. Just based on the density of primes, pk has a probability of being prime approaching 1/(k ln(10)). On this basis, the expected number of primes of the form pk for k less than x is approximately
For x = 500 the expected number of primes is just about 3, whereas we actually have 4. If this formula is anywhere near correct it suggests two things: (1) there are infinitely many primes of the form pk, and (2) we may never know (deterministically) the next prime beyond p38. This is because even to achieve an expected value of 4 requires over x = 5000 digits, which I believe is well beyond the range of practical deterministic primality testing for general numbers (unlike, for instance, Mersenne primes). Here I'm assuming there is no special structure in the decimal expansion of p that could be exploited to test for primality. Of course, we might find another prime pk for some unexpectedly low value of k, but I would venture to say that we will never find two more.
Update: In Dec 2001, Ed T. Prothro reported discovering the next (probable) prime in the decimal digits of p. This took 4.5 months of computation on a 1000 MHz Pentium 3 with Maple running in the background. The probable prime has 16,208 digits:
p
16208 =3141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067
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43936307
Prothro has determined that this number passes the "strong-pseudoprime" test, which means it is almost certainly a prime (although this is not a deterministic proof). In the course of his search he verified that all the numbers from p500 to p16207 are composite.
Based on the probabilistic formula given above, the expected number of primes in the sequence p1, p2... pk reaches 5 for k = 56146, so finding the 5th prime at k = 16208 is a bit earlier than expected, but not much. (The expectation at k=16208 is for 4.46 primes, so it isn't too improbable for the 5th prime to occur at this point.) Using the same formula, the expected number of primes reaches six at k = 561460 digits, so it would be fairly challenging to find the next probable prime in the sequence.