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 p1 = 3 p2 = 31 p6 = 314159 p38 = 31415926535897932384626433832795028841 Maple'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: p16208 = 3141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067 9821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819 6442881097566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127 3724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609 4330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491 2983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940513 2000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923 5420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185 9502445945534690830264252230825334468503526193118817101000313783875288658753320838142061717766914730 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6347160632583064982979551010954183623503030945309733583446283947630477564501500850757894954893139394 4899216125525597701436858943585877526379625597081677643800125436502371412783467926101995585224717220 1777237004178084194239487254068015560359983905489857235467456423905858502167190313952629445543913166 3134530893906204678438778505423939052473136201294769187497519101147231528932677253391814660730008902 7768963114810902209724520759167297007850580717186381054967973100167870850694207092232908070383263453 4520380278609905569001341371823683709919495164896007550493412678764367463849020639640197666855923356 5463913836318574569814719621084108096188460545603903845534372914144651347494078488442377217515433426 0306698831768331001133108690421939031080143784334151370924353013677631084913516156422698475074303297 1674696406665315270353254671126675224605511995818319637637076179919192035795820075956053023462677579 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. Return to MathPages Main Menu