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gd_barnes 2007-07-27 17:35

k's with consecutive small primes
 
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I did some testing on k's with consecutive small primes, i.e. k's that have primes from n=1 thru 6, 1 thru 7, 1-8, etc. up to k=100M. Some of the k's that I found may be excellent candidates for people to search further so I'm listing them here. The ones that are prime for n=1 thru 7 and higher are listed below and I'm attaching a list of the k's that are prime for n=1 thru 6.

Up to k=100M, the following k's are prime for:

n=1 thru 9
42932385

n=1 thru 8
26277285

n=1 thru 7
1082115
1164735
5128905
5154945
7015155
12088065
14421825
18544365
31400085
44087025
47915595
52694085
60627945
69070365
91780965
94708605
95082405

The biggest surprise out of all of this that I found was that k=45 is the only k < 26K that is prime for n=1-5 and is the only k < 200K that is prime for n=1-6!! :surprised

If anyone knows the lowest k that is prime for n=1-10 or would like to do some testing to find it, I'd be interested in seeing the results. I think you'd probably have to test up to k=1G. I was lucky to find one that was prime for n=1-9 out of only 2 candidates that were prime for n=1-8.


Gary

gd_barnes 2007-07-30 23:26

No k's with primes n=1 to 10 up to k=10G
 
An update here...

I extended this search by 100X all the way up to k=10G. There were still no k's that were prime for n=1 thru 10! One big thing that makes 10 consecutive primes harder is the factor of 11. If the k is not divisible by 11 and there's not a 'covering set' of other factors to eliminate it, it occurs (at least) once every 10 n. Over 1/3rd of the k's that were prime for n=2 thru 10 had a factor of 11 for n=1. Obviously the same situation would occur at any number of consecutive primes that is a low prime number minus 1.

I did find one that has 10 consecutive primes though. k=6522452145 is prime for n=2 thru 11. That's a more rare find than one that is prime for n=1 thru 10, so that's a good thing.


Gary


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