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        riddles >> medium >> An all-1 prime number?
        
(Message started by: BNC on Jun 8^th, 2004, 11:46am)

Title: An all-1 prime number?
Post by BNC on Jun 8^th, 2004, 11:46am

Prove that there exist only one prime number between 10 and 10¹⁰ whose digits (in decimal notation) are all "1".

Title: Re: An all-1 prime number?
Post by towr on Jun 8^th, 2004, 1:14pm

Isn't it very,very easy to just factor the 8 possible numbers?

Title: Re: An all-1 prime number?
Post by THUDandBLUNDER on Jun 8^th, 2004, 1:54pm

on 06/08/04 at 13:14:50, towr wrote:

Isn't it very,very easy to just factor the 8 possible numbers?

Perhaps only Euler would agree with you. :P

Title: Re: An all-1 prime number?
Post by Leonid Broukhis on Jun 8^th, 2004, 2:02pm

on 06/08/04 at 13:14:50, towr wrote:

Isn't it very,very easy to just factor the 8 possible numbers?

Actually, one only needs to factor two numbers: 11111 = 41x271 and 1111111 = 239x4649.
The rest are obviously divisible by 11 or 111=3x37.

The first prime of the form 1...1 after 11 is 19 1's, then 23 1's, thanks to the Factorization Engine.

Title: Re: An all-1 prime number?
Post by THUDandBLUNDER on Jun 8^th, 2004, 2:24pm

Quote:

Actually, one only needs to factor two numbers: 11111 = 41x271 and 1111111 = 239x4649.

But towr no doubt knows a 'very, very easy' test for divisibility by 239, even though this puzzle is in Medium. ::)

Title: Re: An all-1 prime number?
Post by Icarus on Jun 8^th, 2004, 6:12pm

I know a quick test to - this is about a 1 minute problem with a computer - 59 secs to hack out a quick script, and 1 sec (rounded up) to run it. 239 doesn't take long to find.

Title: Re: An all-1 prime number?
Post by NickMcG on Jun 8^th, 2004, 7:17pm

For info, the only known repunit primes (where R(x) hax x 1's) are:
R(n) for n = 2, 19, 23, 317, 1031
R(n) for n=49081 and 86453 are probable primes.

Title: Re: An all-1 prime number?
Post by THUDandBLUNDER on Jun 8^th, 2004, 9:16pm

on 06/08/04 at 18:12:59, Icarus wrote:

I know a quick test to - this is about a 1 minute problem with a computer - 59 secs to hack out a quick script, and 1 sec (rounded up) to run it. 239 doesn't take long to find.

I suspect and hope that when BNC here wrote 'Prove' he had another type of script in mind,
and therefore put it in Medium.

Title: Re: An all-1 prime number?
Post by BNC on Jun 8^th, 2004, 10:15pm

Actually, I adopted it from another question, and I'm afraid the adaptation turned it into an easier question that I thought :-[. That should teach me not to attempt adapdations at 2 AM -- everything looks difficult then ;).

As for the factorization thingi -- the original problem would have required a computerless prove that no more than 3 such primes exist in the range (2X :-[).

Title: Re: An all-1 prime number?
Post by Grimbal on Jul 2^nd, 2004, 1:24pm

on 06/08/04 at 11:46:53, BNC wrote:

Prove that there exist only one prime number between 10 and 10¹⁰ whose digits (in decimal notation) are all "1".

I got another one:
Prove that there exist only one prime number between 1 and 10¹⁰ whose digits (in decimal notation) are all "7".
;D

Title: Re: An all-1 prime number?
Post by BNC on Jul 2^nd, 2004, 2:57pm

on 07/02/04 at 13:24:38, Grimbal wrote:

I got another one:
Prove that there exist only one prime number between 1 and 10¹⁰ whose digits (in decimal notation) are all "7".
;D

:P >:( :P