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本帖最后由 愚工688 于 2015-11-19 02:31 编辑
由于大偶数时素对数量的概率计算值的相对误差的波动是很小的,(当然不可能出现某个偶数的相对误差突然为无穷大的情况),因此我们可以通过任意一个偶数的相对误差值,来修正概率计算式,使得其对其它的偶数的素对数量的计算,具有比较高的计算精度。
例:
由素对筛选软件FastGn 得到:G(50000000000) = 79004202,
我的软件的概率计算:Sp( 50000000000 ) ≈ 91406994.34584685 ,
计算相对误差: Δ≈0.156989021240248
取6位小数做修正值,即μ=0.156989,计算450亿-550亿区间偶数的素对数量.
用Sp( m *)=Sp( m )/(1+μ) 来计算450亿-550亿区间偶数的素对数量,这里的μ=0.156989,
G(45000000000) = 143491160 ,Sp( 45000000000 *)≈ 143419976.1 ,Δ≈-0.00049609 ;
G(45000000002) = 55800008 ,Sp( 45000000002 *)≈ 55782342.9 ,Δ≈-0.00031658 ;
G(45000000004) = 55209344 ,Sp( 45000000004 *)≈ 55191427.1 ,Δ≈-0.00032453 ;
G(45000000006) = 117931247 ,Sp( 45000000006 *)≈ 117874583.4 ,Δ≈-0.00048048 ;
G(54900000000) = 175023755 ,Sp( 54900000000 *)≈ 175108272.9 ,Δ≈ 0.00048289 ;
G(54900000002) = 66773893 ,Sp( 54900000002 *)≈ 66802270.3 ,Δ≈ 0.00042498 ;
G(54900000004) = 65129826 ,Sp( 54900000004 *)≈ 65164735 ,Δ≈ 0.00053599 ;
G(54900000006) = 135227059 ,Sp( 54900000006 *)≈ 135291987.1 ,Δ≈ 0.00048014 ;
G(49999999980) = 189678539 ,Sp( 49999999980 *)≈ 189693584.9 ,Δ≈ 0.00007932;
G(49999999982) = 59246939 ,Sp( 49999999982 *)≈ 59253152.9 ,Δ≈ 0.00010488;
G(49999999984) = 65830265 ,Sp( 49999999984 *)≈ 65836836.6 ,Δ≈ 0.00009983;
G(49999999986) = 118502548 ,Sp( 49999999986 *)≈ 118506305.8 ,Δ≈ 0.00003171;
G(49999999988) = 59785070 ,Sp( 49999999988 *)≈ 59786965.1 ,Δ≈ 0.00003170;
G(49999999990) = 84270627 ,Sp( 49999999990 *)≈ 84281178.3 ,Δ≈ 0.00012521;
G(49999999992) = 120389499 ,Sp( 49999999992 *)≈ 120391260.9 ,Δ≈ 0.00001463;
G(49999999994) = 71496593 ,Sp( 49999999994 *)≈ 71500942.2 ,Δ≈ 0.00006083;
G(49999999996) = 59247556 ,Sp( 49999999996 *)≈ 59253152.9 ,Δ≈ 0.00009447;
G(49999999998) = 129296265 ,Sp( 49999999998 *)≈ 129298409.5 ,Δ≈ 0.00001659;
G(50000000000) = 79004202 ,Sp( 50000000000 *)≈ 79004203.9 ,Δ≈ 0.000000024 ,
G(50000000002) = 59262284 ,Sp( 50000000002 *)≈ 59256525.1 ,Δ≈-0.000097176 ,
G(50000000004) = 118490110 ,Sp( 50000000004 *)≈ 118506305.9 ,Δ≈ 0.000136686 ,
G(50000000006) = 68100948 ,Sp( 50000000006 *)≈ 68107072.3 ,Δ≈ 0.000089930 ,
G(50000000008) = 71099519 ,Sp( 50000000008 *)≈ 71103783.5 ,Δ≈ 0.000059979 ,
G(50000000010) = 157988586 ,Sp( 50000000010 *)≈ 158008407.8 ,Δ≈ 0.000125463 ,
G(50000000012) = 65732162 ,Sp( 50000000012 *)≈ 65726186.5 ,Δ≈-0.000090908 ;
G(50000000014) = 61272843 ,Sp( 50000000014 *)≈ 61271185 ,Δ≈-0.000027059 ;
G(50000000016) = 118510495 ,Sp( 50000000016 *)≈ 118516403.8 ,Δ≈ 0.000049859 ;
G(50000000018) = 59292853 ,Sp( 50000000018 *)≈ 59290024.9 ,Δ≈-0.000047697 ;
G(50000000020) = 79010010 ,Sp( 50000000020 *)≈ 79004203.9 ,Δ≈-0.000074386 ;
G(50000000022) = 142186907 ,Sp( 50000000022 *)≈ 142207567.1 ,Δ≈ 0.000145302 ;
G(50000000024) = 70921585 ,Sp( 50000000024 *)≈ 70919098.4 ,Δ≈-0.000035061 ;
G(50000000026) = 59251942 ,Sp( 50000000026 *)≈ 59253153 ,Δ≈ 0.000020438 ;
G(50000000028) = 137457486 ,Sp( 50000000028 *)≈ 137468511.3 ,Δ≈ 0.000080209 ;
G(50000000030) = 79532797 ,Sp( 50000000030 *)≈ 79541647.5 ,Δ≈ 0.000111281 ;
G(50000000032) = 59282642 ,Sp( 50000000032 *)≈ 59293820.9 ,Δ≈ 0.000188570 ;
G(50000000034) = 118500487 ,Sp( 50000000034 *)≈ 118506305.9 ,Δ≈ 0.000049104 ;
G(50000000036) = 74548291 ,Sp( 50000000036 *)≈ 74548119.7 ,Δ≈-0.000002298 ;
G(50000000038) = 59294346 , Sp( 50000000038 *)≈ 59296371.9 ,Δ≈ 0.000034167 ;
G(50000000040) = 159496823 ,Sp( 50000000040 *)≈ 159513249.9 ,Δ≈ 0.000102990 ;
我说的概率计算值的相对误差的变化规律性,通过具体的偶数的素对数量的计算值以及相对误差值,得到了一致的结果。 |
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