January 2nd, 2006

Power Floor Primes

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This topic was brought to my attention during a conversation with Prof. Wiesenbauer of the Technical University of Vienna. He calculated the results below, which I could confirm totally independend.

For a given n, find the longest sequence of primes qn,k = floor( ( pn + e )k ), where pn is the n-th prime number and e is a number in [0,1[. The lengths of these sequences build a series an = { 8, 7, 8, 5, 10, 12, 16, 14, 18, 22, 24, 26, 27, 28, 34, 35, 37, 39, 40, 45, 43, 46, 49, 51, 55, 57, ... }.
The calculation of the longest sequences for a given n is straight forward. Start considering e in the set [0,1[ and k = 1. Now systematically increase k by 1 and remove these parts of the set where int( ( pn + e ) k ) is not prime. The set becomes progressively thinner until it vanishes. The last value of k before the set vanishes, is the length of the longest possible sequence. It is possible that there is more than one longest sequence, i.e. that the last set before vanishing is the union of disjunct subsets.

The following table gives the length of the longest sequence, the number of disjunct subsets before vanishing, the total length of these subsets, the calculation time on my P4/3.2GHz and a heuristic estimation of the length of the longest sequence deduced from the logarithmic integral estimation for the prime counting function p(n).

n pn an # of subsets total length P4/3.2GHz time in sec. est. with li(x)
1 2 8 1 3,051 10-4 < 1 5
2 3 7 1 6,228 10-5 < 1 5
3 5 8 1 3,100 10-7 < 1 6
4 7 5 3 8,857 10-5 < 1 7
5 11 10 1 2,894 10-11 < 1 10
6 13 12 1 2,222 10-14 < 1 11
7 17 16 1 1,370 10-20 < 1 14
8 19 14 4 4,230 10-18 < 1 15
9 23 18 2 6,176 10-25 < 1 17
10 29 22 1 7,219 10-33 1 21
11 31 24 1 1,035 10-36 2 22
12 37 26 1 1,819 10-41 7 25
13 41 27 3 1,062 10-43 21 27
14 43 28 4 8,617 10-46 32 28
15 47 34 1 3,732 10-59 111 30
16 53 35 1 3,715 10-61 322 33
17 59 37 2 6,576 10-66 1.361 36
18 61 39 1 2,739 10-70 1.876 37
19 67 40 4 4,768 10-73 7.528 40
20 71 45 1 6,536 10-84 18.018 42
21 73 43 3 2,386 10-80 31.307 43
22 79 46 2 1,217 10-87 140.524 46
23 83 49 2 2,353 10-94 253.766(1) 48
24 89 51 1 5,038 10-100 689.034(1) 51
25 97 55 4 2,712 10-109 9.548.897(2) 54
26 101 57 1 7,895 10-115 23.602.298(2) 56
27 103 ? ? ? - 57
28 107 ? ? ? - 59
29 109 ? ? ? - 60
30 113 ? ? ? - 62

(1) extrapolated times proportionally to clock rate
(2) done with a shared network of about 25 CPU's with average performance of 2,4GHz.

The intervall for n = 26 which produces a sequence of 57 primes is [a1/57, (a+1)1/57[ with
a = 2253991540629602344107433341245279404398438508832008004810967711479085968783190844558218891731843032605525294461441
The smallest fraction p / q with p < q so that floor( ( 101 + p / q )k ) is prime for k = 1(1)57 is
p / q = 429839253817241228480909524515793682631599571333522464544 / 985853861389471106432485444190026022526545502209436635483 = 0,436007...


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