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Swiss primes: This plot shows what I am calling Swiss prime numbers up to 30 million, shown in magenta open circles. I am sure there is a mathematical definition for these prime numbers but I haven't found it yet. They are the prime numbers that are equidistant from their two neighboring primes. In this case, the "loneliness," defined as sqrt(distance_lower * distance_upper) (the geometric mean of separations) is an integer and is equal to the distance from the lower or upper neighboring prime. As you can see from the magenta plot, with a logarithmic x-axis, the loneliness of the Swiss primes also grows faster than log(prime number), as shown by the positive curvature of the envelope of the magneta circles. If you know the proper name for these "Swiss primes" please let me know! See also my previous post: https://plus.google.com/110146523961072500429/posts/TsSWSprcYaH
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Steve Koch's profile photoGary Johnson's profile photo
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Thank you to former Junior Lab student John Callow (now UC-Riverside Math) for following comments: "Well I found this http://en.wikipedia.org/wiki/Balanced_prime . These are the types of primes you're looking at right?" ... "As for the greatest distance between two primes I believe for any value N you can find two neighboring primes such that their distance from each other is >N. I vaguely remember proving something like this in an undergrad algebra course or something."
 
OK, so they are balanced primes and their maximum value appears to grow faster than log(Pn). Too bad I'm not a mathematician, that's about all I can add to the discussion with myself right now. Well, also: there are 57,243 balanced primes <= 30,000,000. Balanced Prime #57,243 is 29,998,259, which is 12 away from neighboring primes. (see http://www.numberempire.com/29998259)
 
That above link to 29998259 on number empire (amazing by the way) says that it is the 1,857,756th prime. I found that it is the 57,243th balanced prime. So, the fraction of primes <30million that are balanced is 0.0308 or about 3%.
 
the fraction of numbers <30million that are prime is 0.0619 or about 6% (1857859/30000000). And the fraction of numbers <=30million that are balanced primes is 0.0019 or 0.1%. That seems like a lot.
 
The maximum isolation of a balanced prime less than 30,000,000 is 72 (=2*2*2*3*3), which occurs 4 times: 23,346,809; 25,443,361; 28,933,939; and 29,747,071. For 23,346,809, 72 is 4.24 times its natural log.
 
Besides the balanced prime 5 (between 3 and 7), the isolation or loneliness of a balanced prime appears to have to be a multiple of 6 (see graph above with magenta at 2, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72). That must mean something.
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