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Topic: An all-1 prime number? (Read 901 times) |
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BNC
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An all-1 prime number?
« on: Jun 8th, 2004, 11:46am » |
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Prove that there exist only one prime number between 10 and 1010 whose digits (in decimal notation) are all "1".
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towr
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Re: An all-1 prime number?
« Reply #1 on: Jun 8th, 2004, 1:14pm » |
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Isn't it very,very easy to just factor the 8 possible numbers?
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ThudnBlunder
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Re: An all-1 prime number?
« Reply #2 on: Jun 8th, 2004, 1:54pm » |
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on Jun 8th, 2004, 1:14pm, towr wrote:| Isn't it very,very easy to just factor the 8 possible numbers? |
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Perhaps only Euler would agree with you. 
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Leo Broukhis
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Re: An all-1 prime number?
« Reply #3 on: Jun 8th, 2004, 2:02pm » |
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on Jun 8th, 2004, 1:14pm, towr wrote:| Isn't it very,very easy to just factor the 8 possible numbers? |
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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.
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ThudnBlunder
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Re: An all-1 prime number?
« Reply #4 on: Jun 8th, 2004, 2:24pm » |
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Quote:| Actually, one only needs to factor two numbers: 11111 = 41x271 and 1111111 = 239x4649. |
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But towr no doubt knows a 'very, very easy' test for divisibility by 239, even though this puzzle is in Medium. 
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| « Last Edit: Jul 2nd, 2004, 3:16pm by ThudnBlunder » |
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Icarus
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Re: An all-1 prime number?
« Reply #5 on: Jun 8th, 2004, 6:12pm » |
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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.
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NickMcG
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Re: An all-1 prime number?
« Reply #6 on: Jun 8th, 2004, 7:17pm » |
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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.
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| « Last Edit: Jun 8th, 2004, 7:18pm by NickMcG » |
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ThudnBlunder
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Re: An all-1 prime number?
« Reply #7 on: Jun 8th, 2004, 9:16pm » |
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on Jun 8th, 2004, 6:12pm, 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. |
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I suspect and hope that when BNC here wrote 'Prove' he had another type of script in mind,
and therefore put it in Medium.
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| « Last Edit: Jun 8th, 2004, 9:19pm by ThudnBlunder » |
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BNC
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Re: An all-1 prime number?
« Reply #8 on: Jun 8th, 2004, 10:15pm » |
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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 ).
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Grimbal
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Re: An all-1 prime number?
« Reply #9 on: Jul 2nd, 2004, 1:24pm » |
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on Jun 8th, 2004, 11:46am, BNC wrote:| Prove that there exist only one prime number between 10 and 1010 whose digits (in decimal notation) are all "1". |
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I got another one:
Prove that there exist only one prime number between 1 and 1010 whose digits (in decimal notation) are all "7".
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BNC
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Re: An all-1 prime number?
« Reply #10 on: Jul 2nd, 2004, 2:57pm » |
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on Jul 2nd, 2004, 1:24pm, 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".
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