[math-fun] Primes in First N Digits of Pi
These are prime: 3 31 314159 31415926535897932384626433832795028841 (first 38 digits of Pi). The first 16208 digits of Pi are a probable prime. Ed T. Prothro found this in 2001. I just finished checking up through 40000 digits. There are no other numbers in this sequence for which Mathematica's PrimeQ[ ] is TRUE. Has anyone checked beyond this limit? Bob Baillie
At 9:16 AM -0400 6/23/04, Robert Baillie wrote:
These are prime: 3 31 314159 31415926535897932384626433832795028841 (first 38 digits of Pi).
The first 16208 digits of Pi are a probable prime. Ed T. Prothro found this in 2001.
I just finished checking up through 40000 digits. There are no other numbers in this sequence for which Mathematica's PrimeQ[ ] is TRUE.
Has anyone checked beyond this limit?
Could you also try digits of pi to other bases? Particularly base 2. Paul
=Paul R. Pudaite
=Robert Baillie The first 16208 digits of Pi are a probable prime. Could you also try digits of pi to other bases? Particularly base 2.
While we're asking: what about numbers *all* of whose initial strings are primes? Are there any? Infinitely many? In binary: obviously none since you can't even start with 0 or 1. In ternary: each choice is forced, and you can only get as far as "2122" = 71. If such exist: What is the smallest base? Is there a sharp lower bound on the number itself? If not: How far can you get in each base? (Nice sequence for OEIS, maybe?)
On Wed, 23 Jun 2004, Marc LeBrun wrote:
While we're asking: what about numbers *all* of whose initial strings are primes?
Are there any? Infinitely many?
Yes, there are just a finite number of both types (all initial strings are prime or all terminal strings are prime). See http://mathworld.wolfram.com/TruncatablePrime.html Just about any question of this type that one can ask has been examined by prime enthusiasts. The only problem is finding out what they have been named. In this case it is left or right truncatable prime and undoubtedly others have looked at them and given them other names. I heard about these primes from John Maxfield back in 1966 or so, but had a problem locating this reference. Finally I found it by going to mathworld and looking thru all entries returned by a search on primes. Speaking of truncation, the MathWorld entry mentions a similar class of primes called "Henry VIII primes".:-) --Edwin
While we're asking: what about numbers *all* of whose initial strings are primes? In binary: obviously none since you can't even start with 0 or 1. In ternary: each choice is forced, and you can only get as far as "2122" = 71. How far can you get in each base?
Since you asked... (For large digits: a=10, b=11, ..., z=35.) base how far you can get digits in base ten ---- ------------------------- ------ -------------------------------------- 3 2122 4 71 4 2333 4 191 5 34222 5 2437 6 2155555 7 108863 7 25642 5 6841 8 21117717 8 4497359 9 3444224222 10 1355840309 10 73939133 8 73939133 11 29668286aa 10 6774006887 12 375bb5b515 10 18704078369 13 b6c2ca8a8a 10 122311273757 14 2dd35b9d399395b3d 17 6525460043032393259 15 72424e42eee8e 13 927920056668659 16 3b9bf319bd51ff 14 16778492037124607 17 5g4cee8ec688cac86g 18 4928397730238375565449 18 dh17hb7bbd75bdb 15 5228233855704101657 19 3ec8gi8gicieg8c 15 3013357583408354653 20 23hbh9d19hh9jddj9 17 1437849529085279949589 21 3824a4gga4ag82kka8 18 101721177350595997080671 22 5h975fflljf3hl3f33f3 20 185720479816277907890970001 23 dek6icce8ee2k26 15 158208158913013692383 24 3b5j511h5njnn55b7jdbnn7h 24 192747244030905257036482742599289 25 jcmiiieiioic4eigo2 18 11360039924980123824119977 26 hj1fhn97jf9p7pffj19 19 522764314648992960422987767 27 2dmmkqemam4884qmaeag2 21 106521223483392113109841556843 28 5953r9jhj5pff3r3h3d9n 21 467437774672035454997088263971 29 3k6qoo6682o4ag4gg6q82c 22 18980691336146397055451904000521 30 jnhj77ddnt7thdd177hd7b 22 206971354022501468249535515240921 31 jc642uis2s8goquskmii2a 22 403878995374635723531460715056361 32 7ht59vf3pdrrj7pd3371rb5 23 9813093725765026702961210138094949 33 3wek8qagqw8gw4e4kwgeaa2 23 10174889780995609522983172669668593 34 35x5fpf5r7xbxd9lrb1brxxvt 25 18085876810004448001794542893991790487 35 t6cgg4g68i4mc26gcoyycwcc 24 9520817609816167868579578513867491007 36 dzjzjpddp7j55znppz71pd7h 24 8723727825272063982605771015871962141 -- Don Reble djr@nk.ca
Hey, that's cool. Surely this was already known, but not to me: naively, you expect infinitely many primes of that form. (Pretending that pi is a string of random digits and other such falsities...) The number of primes less than n is about n/ln n, so the number of primes between b^k and b^(k+1) is about (b^k / ln b) * (b/(k+1)-1/k), and the probability that a randomly-chosed number between b^k and b^(k+1) is prime simplifies to pr_k \approx (b/(k+1)-1/k) / (b-1) ln b Now we just sum this from k=0 to infinity. There's a base b constant factor of 1/(b-1) ln b, and we have to sum b/(k+1)-1/k. But of course 1/(k+1)-1/k is a telescoping sum, and the remainder is the sum of (b-1)/(k+1), which diverges. (I guess we can do this more sloppily and just observe that the sum of 1/ln(b^k) diverges.) So, to be precise, I'm claiming that if you take a random set consisting of one number with each of 1,2,3,4,... digits in base b, you expect it to contain infinitely many primes. I can't think of any reason that the numbers' digits all being initial subsequences of the same infinite sequence should seriously affect this, nor of any reason that pi should be non-generic in this context. Heh. Good luck finding the next one. --Michael Kleber kleber@brandeis.edu On Jun 23, 2004, at 9:16 AM, Robert Baillie wrote:
These are prime: 3 31 314159 31415926535897932384626433832795028841 (first 38 digits of Pi).
The first 16208 digits of Pi are a probable prime. Ed T. Prothro found this in 2001.
I just finished checking up through 40000 digits. There are no other numbers in this sequence for which Mathematica's PrimeQ[ ] is TRUE.
Has anyone checked beyond this limit?
Bob Baillie
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participants (6)
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Don Reble -
Edwin Clark -
Marc LeBrun -
Michael Kleber -
Paul R. Pudaite -
Robert Baillie