Sunday, 10 June 2012

Method for the reduction of errors in the computation, the number of Prime numbers in an interval in several times

The formula for calculation of the value of (m)
For the application of the method of reduction of errors in the computation, draw up a table of values in the formulas of the algorithms sieve of Eratosthenes up to the value of (p_n). That is not a problem, the table is compiled once and then is used in calculations. For example: the First column is the value of (n). The second column value (p_n). The third column, the value calculated according to the formula algorithm sieve of Eratosthenes.


1
2
0,5
2
3
0,33333333333333333333333333333333
3
5
0,26666666666666666666666666666667
4
7
0,22857142857142857142857142857143
5
11
0,20779220779220779220779220779221




By value (m) find the value (p_n^/)
 You can begin to reverse отсчет. this formula







find the number of composite numbers in the interval (0.m), which at the same time is the number of Prime numbers in an interval

 

For example: p_n=691 m=44481,2382446949 √m=210,9057567841497 199<210,9057567841497>211
(p_n^/)=199
477481* 0,0852192648921898=40690,5791998768 (39809) - the Number of Prime numbers in parentheses table value.

E=40690,57981998768-39809=881,5798199876779 the error in the calculation of the basic formula.

 Calculation by the method of descent and reverse
0477481*,0852192648921898 / 0,9147807351078102*0,8961053609991042=39859,87605494952
E=39859,87605494952-39809=50,87605494952096 Error of calculation by the method of descent and reverse.
.Here you have the result of reducing errors in the computation of the few times


Another example:
(p_n)=683
0,085342771073193
39811,46393516273 (38826) E=985,46393516273
(m)= 43526,10210261459
√m=208,6291017634275 (p_n^/=199)
0,10389463900089585434036970613043
43526,10210261459*0,8961053609991042=39003,97343754732 (38826) E=177,9734375473155
Reduction of errors in the computation of several times.

 

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