Prime-Digit Primes...

petrw1 · 2009-12-14, 22:48

Not sure if this puzzle is new or even interesting...

Given there is NO largest prime number but is there a largest prime number such that each and every digit is also prime (i.e. 2, 3, 5, or 7)? OR what is the largest known such prime?

I found:
- Sloane A124888 Primes with prime number of only prime digits (i.e. 2,3,5,7). The largest listed there is 27277.
- Sloane A152427 Primes with prime digits - However it includes the digit 1(?)

wblipp · 2009-12-15, 00:02

There should be LOTS of these. You can use 2,3,5, or 7, so among 100 digit numbers there are 4^100 that use only prime digits. We expect about 1 in 230 (ln(10^100)) to be prime - the expected number grows with each decade.

William

CRGreathouse · 2009-12-15, 03:13

The largest hundred-digit member of the sequence is 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777573553. The largest thousand-digit member is $\frac79\left(10^{1000}-1\right)-44054$ .

I'm slightly more optimistic than wblipp; I expect there are about 8.7616 × 10⁵⁷ 100-digit members of the sequence.

petrw1 · 2009-12-15, 22:23

Hmmm... I didn't expect so many.

I did a quick and dirty test of the first 50 Million primes; partially counting and partially estimating. I figure about half a percent qualify or about 250,000.

CRGreathouse · 2009-12-16, 02:45

Quote:

Originally Posted by petrw1

I did a quick and dirty test of the first 50 Million primes; partially counting and partially estimating.

Hmm, the primes up to 982451653 -- the 1-, 2-, ..., and 8-digit primes, plus most of the 9-digit primes. I'll use this below.

Quote:

Originally Posted by petrw1

I figure about half a percent qualify or about 250,000.

The simple heuristic (4/log 10 + 4^2/log 100 + ... + 4^9/log 1000000000) predicts 17,651.
My heuristic (basically the same, but dividing by log - 1 and residues mod 10) predicts there are 23,254.

I counted 23,169 with Pari:

Code:

est(n)=4^n*5/4/(log(10)*n-1)
sum(i=1,9,est(i))
sum(i=1,9,4^i/(i*log(10)))
works(n)=while(n>9,if(isprime(n%10),n\=10,return(0)));isprime(n)
t=0;forprime(p=2,982451653,t+=works(p));t

in 62 seconds. (Also, Pari took ~2 seconds to build the prime list.) Faster code could be written with BCDs and a shift table for prime final digits -- but this is good enough for me.

petrw1 · 2009-12-16, 04:34

Quote:

Originally Posted by petrw1

I did a quick and dirty test of the first 50 Million primes; partially counting and partially estimating. I figure about half a percent qualify or about 250,000.

Check that not 0.5% but 0.05% or 25,000....now we are closer.

I counted a few ranges and found that about 1,000 out of each 1,000,000 passed (0.1%) but that about half the ranges had none i.e. ranges that all the primes started with 4, 6, 8 or 9 or had 0,1,4,6,8,9 as the second digit in the entire range.

Thanks

R.D. Silverman · 2009-12-16, 18:47

Quote:

Originally Posted by petrw1

Not sure if this puzzle is new or even interesting...

Given there is NO largest prime number but is there a largest prime number such that each and every digit is also prime (i.e. 2, 3, 5, or 7)? OR what is the largest known such prime?

Although a proof is lacking, strong heuristics exist that suggest
[as with Mersenne primes] that there should be infinitely many such
primes.

The following, however, can be proven:

While the sum of the reciprocals of all the primes diverges,
the sum of the reciprocals of all primes containing just prime digits
is convergent.

CRGreathouse · 2009-12-16, 19:36

Quote:

Originally Posted by R.D. Silverman

The following, however, can be proven:

While the sum of the reciprocals of all the primes diverges,
the sum of the reciprocals of all primes containing just prime digits
is convergent.

Of course the sum of the reciprocals of the *integers* containing just prime digits is convergent:
$\sum_{n\in\mathscr{PD}}1/n$
$=\sum_{d=1}^\infty\sum_{10^{d-1}\le n<10^n,n\in\mathscr{PD}}1/n$
$<\sum_{d=1}^\infty4^d\cdot1/10^{d-1}$
$=10\sum_{d=1}^\infty.4^d=20/3$

Jens K Andersen · 2009-12-16, 21:18

Quote:

Originally Posted by petrw1

what is the largest known such prime?

The largest known is (10^40950+1)*(10^20055+1)*(10^10374+1)*(10^4955+1)*(10^2507+1)*(10^1261+1)*(3*R(1898)+555531001*10^940-R(958))+1.
It has 82000 digits and was found by David Broadhurst in 2003: http://tech.groups.yahoo.com/group/p...m/message/3846. Remarkably it is a proven prime and was number 486 at the time: http://primes.utm.edu/primes/page.php?id=66729.

Mini-Geek · 2009-12-16, 21:30

Quote:

Originally Posted by Jens K Andersen

...Remarkably it is a proven prime and was number 486 at the time: http://primes.utm.edu/primes/page.php?id=66729.

No it's not. From that link:

Verification status: PRP

The largest ECPP proof on the prime pages is 20562 digits. http://primes.utm.edu/top20/page.php?id=27

Jens K Andersen · 2009-12-16, 21:58

Quote:

Originally Posted by Jens K Andersen

It has 82000 digits and was found by David Broadhurst in 2003: http://tech.groups.yahoo.com/group/p...m/message/3846. Remarkably it is a proven prime

Quote:

Originally Posted by Mini-Geek

No it's not. From that link:

Verification status: PRP

The largest ECPP proof on the prime pages is 20562 digits. http://primes.utm.edu/top20/page.php?id=27

Yes, it is proven with no need for ECPP above 4560 digits. Details of David's clever Konyagin-Pomerance proof with 30.18% factorization of N-1 are at the first link. The Prime Pages doesn't repeat some of the more complicated proofs. "Verification status: PRP" means that The Prime Pages only made a PRP test in this case. But Chris Caldwell reads the primeform list where the prime was announced, and David is trusted to make complicated proofs. His Prime Pages accounts have many accepted proofs saying "Verification status: PRP".
By the way, the 20562-digit ECPP proof you meantion also says "Verification status: PRP": http://primes.utm.edu/primes/page.php?id=77907.

2009-12-14, 22:48	#1
petrw1 1976 Toyota Corona years forever! "Wayne" Nov 2006 Saskatchewan, Canada 2²×7×167 Posts	Prime-Digit Primes... Not sure if this puzzle is new or even interesting... Given there is NO largest prime number but is there a largest prime number such that each and every digit is also prime (i.e. 2, 3, 5, or 7)? OR what is the largest known such prime? I found: - Sloane A124888 Primes with prime number of only prime digits (i.e. 2,3,5,7). The largest listed there is 27277. - Sloane A152427 Primes with prime digits - However it includes the digit 1(?) Last fiddled with by petrw1 on 2009-12-14 at 22:48

2009-12-15, 00:02	#2
wblipp "William" May 2003 New Haven 2·7·13² Posts	There should be LOTS of these. You can use 2,3,5, or 7, so among 100 digit numbers there are 4^100 that use only prime digits. We expect about 1 in 230 (ln(10^100)) to be prime - the expected number grows with each decade. William Last fiddled with by wblipp on 2009-12-15 at 00:03

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2009-12-15, 03:13	#3
CRGreathouse Aug 2006 3·1,993 Posts	The largest hundred-digit member of the sequence is 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777573553. The largest thousand-digit member is $\frac79\left(10^{1000}-1\right)-44054$ . I'm slightly more optimistic than wblipp; I expect there are about 8.7616 × 10⁵⁷ 100-digit members of the sequence.

2009-12-15, 22:23	#4
petrw1 1976 Toyota Corona years forever! "Wayne" Nov 2006 Saskatchewan, Canada 1001001000100₂ Posts	Hmmm... I didn't expect so many. I did a quick and dirty test of the first 50 Million primes; partially counting and partially estimating. I figure about half a percent qualify or about 250,000.