[math-fun] Prime factors of 10^N + 1
On 2017-09-25 15:36, Dan Asimov wrote:
Decided to list these just for the heck of it. When's the next prime after 101? Mathematica says its ≥ 10^2^20+1. --rwg
Some amusing almost-patterns for 10^0 + 1 through 10^19 + 1:
1 = 1
11 = 11
101 = 101
1001 = 7 * 11 * 13
10001 = 73 * 137
100001 = 11 * 9091
1000001 = 101 * 9901
10000001 = 11 * 909091
100000001 = 17 * 5882353
1000000001 = 7 * 11 * 13 * 19 * 52579
10000000001 = 101 * 3541 * 27961
100000000001 = 11 * 11 * 23 * 4093 * 8779
1000000000001 = 73 * 137 * 99990001
10000000000001 = 11 * 859 * 1058313049
100000000000001 = 29 * 101 * 281 * 121499449
1000000000000001 = 7 * 11 * 13 * 211 * 241 * 2161 * 9091
10000000000000001 = 353 * 449 * 641 * 1409 * 69857
100000000000000001 = 11 * 103 * 4013 * 21993833369
1000000000000000001 = 101*9901*999999000001
10000000000000000001 = 11*909090909090909091
10^20 + 1 is too large for my software.
—Dan
Not what you asked, but (10^n + 1) / 11 is prime for n = 5, 7, 19, 31, 53, 67, 293, 641 On Tue, Sep 26, 2017 at 11:22 PM, Bill Gosper <billgosper@gmail.com> wrote:
On 2017-09-25 15:36, Dan Asimov wrote:
Decided to list these just for the heck of it. When's the next prime after 101? Mathematica says its ≥ 10^2^20+1. --rwg
Some amusing almost-patterns for 10^0 + 1 through 10^19 + 1:
1 = 1
11 = 11
101 = 101
1001 = 7 * 11 * 13
10001 = 73 * 137
100001 = 11 * 9091
1000001 = 101 * 9901
10000001 = 11 * 909091
100000001 = 17 * 5882353
1000000001 = 7 * 11 * 13 * 19 * 52579
10000000001 = 101 * 3541 * 27961
100000000001 = 11 * 11 * 23 * 4093 * 8779
1000000000001 = 73 * 137 * 99990001
10000000000001 = 11 * 859 * 1058313049
100000000000001 = 29 * 101 * 281 * 121499449
1000000000000001 = 7 * 11 * 13 * 211 * 241 * 2161 * 9091
10000000000000001 = 353 * 449 * 641 * 1409 * 69857
100000000000000001 = 11 * 103 * 4013 * 21993833369
1000000000000000001 = 101*9901*999999000001
10000000000000000001 = 11*909090909090909091
10^20 + 1 is too large for my software.
—Dan
math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
that's https://oeis.org/A001562 : 5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207 Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Wed, Sep 27, 2017 at 12:47 AM, James Buddenhagen <jbuddenh@gmail.com> wrote:
Not what you asked, but (10^n + 1) / 11 is prime for n = 5, 7, 19, 31, 53, 67, 293, 641
On Tue, Sep 26, 2017 at 11:22 PM, Bill Gosper <billgosper@gmail.com> wrote:
On 2017-09-25 15:36, Dan Asimov wrote:
Decided to list these just for the heck of it. When's the next prime after 101? Mathematica says its ≥ 10^2^20+1. --rwg
Some amusing almost-patterns for 10^0 + 1 through 10^19 + 1:
1 = 1
11 = 11
101 = 101
1001 = 7 * 11 * 13
10001 = 73 * 137
100001 = 11 * 9091
1000001 = 101 * 9901
10000001 = 11 * 909091
100000001 = 17 * 5882353
1000000001 = 7 * 11 * 13 * 19 * 52579
10000000001 = 101 * 3541 * 27961
100000000001 = 11 * 11 * 23 * 4093 * 8779
1000000000001 = 73 * 137 * 99990001
10000000000001 = 11 * 859 * 1058313049
100000000000001 = 29 * 101 * 281 * 121499449
1000000000000001 = 7 * 11 * 13 * 211 * 241 * 2161 * 9091
10000000000000001 = 353 * 449 * 641 * 1409 * 69857
100000000000000001 = 11 * 103 * 4013 * 21993833369
1000000000000000001 = 101*9901*999999000001
10000000000000000001 = 11*909090909090909091
10^20 + 1 is too large for my software.
—Dan
math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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If on odd factor F divides N, then 10^(N/F) + 1 is a divisor of 10^N + 1. 3 divides 6, so 10^2+1 = 101 divides 10^6+1 = 1000001. Corollary: If 10^N+1 is prime, N has no odd factors >1. Corollary: If 10^N+1 is prime, N is a power of 2. Heuristic: The chance that N is prime is about 1/logN. Caveat: This needs to be adjusted for N being non-random (e.g. odd). Consequence: The expected number of primes in any sequence that grows much faster than exponentially is finite. Empirically, no formula for N has been found that raises the probability much above a constant K times 1/logN. So there might be an infinity of Mersenne primes 2^N-1, but we probably know all the Fermat primes 2^2^N+1. There may be an infinity of repunit primes (111...111), but we probably know all the 10^N+1 primes: 2, 11, and 101. Rich ----- Quoting Bill Gosper <billgosper@gmail.com>:
On 2017-09-25 15:36, Dan Asimov wrote:
Decided to list these just for the heck of it. When's the next prime after 101? Mathematica says its ? 10^2^20+1. --rwg
Some amusing almost-patterns for 10^0 + 1 through 10^19 + 1:
1 = 1
11 = 11
101 = 101
1001 = 7 * 11 * 13
10001 = 73 * 137
100001 = 11 * 9091
1000001 = 101 * 9901
10000001 = 11 * 909091
100000001 = 17 * 5882353
1000000001 = 7 * 11 * 13 * 19 * 52579
10000000001 = 101 * 3541 * 27961
100000000001 = 11 * 11 * 23 * 4093 * 8779
1000000000001 = 73 * 137 * 99990001
10000000000001 = 11 * 859 * 1058313049
100000000000001 = 29 * 101 * 281 * 121499449
1000000000000001 = 7 * 11 * 13 * 211 * 241 * 2161 * 9091
10000000000000001 = 353 * 449 * 641 * 1409 * 69857
100000000000000001 = 11 * 103 * 4013 * 21993833369
1000000000000000001 = 101*9901*999999000001
10000000000000000001 = 11*909090909090909091
10^20 + 1 is too large for my software.
?Dan
math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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