October 1st 2005## Admissible Prime Sets

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## 3. Counter-example

Finding a counter-example for the second Hardy-Littlewood conjecture, i.e. the prime counting function is not convex, is more or less impossible. However, I did some estimates and calculations related to this issue:

The only set for

- If there is a counter-example based on one of the 12 constellations with length 3159 it should be somewhere around 10
^{1183}. Every counter-example based on a set with larger length is with the utmost probability bigger. This result can be achieved with some calculation under the assumption of the heuristically quantified Dickson's conjecture.- Let
Pbe the product of the first_{m}kprime numbers and 0 =a_{1}<a_{2}< ... <a_{447}= 3158 one of the 12 admissible sets with lengthn= 3159 and sizek= 447. With the Chinese Remainder Theorem it is possible to find all numbersN<P, so that non of the 447 numbers_{m}N+ahas a divisor smaller or equal_{i}p_{k}. The following table gives the maximum number of primes in the sets {N+a}_{i}_{i=1(1)447, N < Pm, p | (N + ai) <=> p > pm}and the length of the longest sequence of primesN,N+a,_{2}N+a, ... at the beginning of such a set._{3}

max number longest sequence total number mp_{m}P_{m}of primes of primes of possible N24 89 2,4 10 ^{34}53 0 2 25 97 2,3 10 ^{36}49 1 12 26 101 2,3 10 ^{38}48 1 12 27 103 2,4 10 ^{40}48 1 12 28 107 2,6 10 ^{42}53 1 12 29 109 2,8 10 ^{44}46 1 12 30 113 3,2 10 ^{46}43 1 12 31 127 4,0 10 ^{48}46 1 12 32 131 5,3 10 ^{50}42 0 12 33 137 7,2 10 ^{52}48 0 12 34 139 1,0 10 ^{55}44 2 24 35 149 1,5 10 ^{57}44 0 24 36 151 2,3 10 ^{59}40 2 24 37 157 3,5 10 ^{61}46 1 72 38 163 5,8 10 ^{63}39 1 96 39 167 9,6 10 ^{65}41 3 288 40 173 1,7 10 ^{68}42 2 900 41 179 3,0 10 ^{70}44 3 4.716 42 181 5,4 10 ^{72}47 3 18.864 43 191 1,0 10 ^{75}50 4 188.640 44 193 2,0 10 ^{77}50 5 943.200 45 197 3,9 10 ^{79}51 5 6.602.400 46 199 7,8 10 ^{81}54 6 41.126.400 47 211 1,6 10 ^{84}? 7 435.456.000

m= 46 which has 54 primes is560651168351945542265568623882043856206800351003857634040218867197345773820708901 + Set 4The only set for

m= 47 which starts with 7 primes is a subset of Set 11677039857508525000692405150755887372006330981606535979970170582458895880942908768091 + {0, 6, 8, 20, 30, 32, 36}main page - page 1 - page 2 - page 3