抛砖引玉

yangchuanju

512942430的哥猜素数对数
真值：4123665
分解式：512942430=2 * 3 * 5 * 7 * 11 * 13 * 19 * 29 * 31
平方根：22648.232
连乘积计算值
512942430/4*(2*4/3*6/5*10/9*12/11*18/17*28/27*30/29)*∏(p-2)/p
=128235607.5*4.40592537955621*0.0082673446967518=4671020.96385265
连乘积计算值比真值大13.27%。

哈代计算值
0.660161816*(2*4/3*6/5*10/9*12/11*18/17*28/27*30/29)*512942430/ln(512942430)^2
=0.660161816*4.40592537955621*512942430/402.230066598337=3709211.79789398
哈代计算值比真值小10.05%。

请愚公老师根据您的公式计算计算，看一看误差有多大！

愚工688

我用下界计算式计算的偶数素数对下界值，相对误差都是比较小的：
重新把连续偶数的全部下界计算值的相对误差计算出来，以便大家观察一下素对下界计算值的相对误差是多么的稳定，
而区域下界值,infS(m)则是线性向上的（最大素数不变的区域内）




G(512942430) = 4123665；inf( 512942430 )≈  4117250.7 , Δ≈-0.00156,infS(m) = 934480.35 , k(m)=
G(512942432) = 936995  ；inf( 512942432 )≈  934480.4   , Δ≈-0.00268,infS(m) = 934480.35 , k(m)=
G(512942434) = 936375  ；inf( 512942434 )≈  934480.4   , Δ≈-0.00202,infS(m) = 934480.36 , k(m)=
G(512942436) = 1873378；inf( 512942436 )≈  1868960.7 , Δ≈-0.00236,infS(m) = 934480.36 , k(m)=
G(512942438) = 945310  ；inf( 512942438 )≈  943919.6   , Δ≈-0.00147,infS(m) = 934480.37 , k(m)=
G(512942440) = 1249865；inf( 512942440 )≈  1246322    , Δ≈-0.00283,infS(m) = 934480.37 , k(m)=
G(512942442) = 1924038；inf( 512942442 )≈  1919644.2 , Δ≈-0.00228,infS(m) = 934480.37 , k(m)=
G(512942444) = 1123885；inf( 512942444 )≈  1121376.5 , Δ≈-0.00223,infS(m) = 934480.38 , k(m)= 1.2
G(512942446) = 946224  ；inf( 512942446 )≈  943213.8   , Δ≈-0.00318,infS(m) = 934480.38 , k(m)=  
G(512942448) = 1873265；inf( 512942448 )≈  1868960.8 , Δ≈-0.00230,infS(m) = 934480.38 , k(m)= 2
G(512942450) = 1248651；inf( 512942450 )≈  1245973.9 , Δ≈-0.00214,infS(m) = 934480.39 , k(m)=  
G(512942452) = 1040717；inf( 512942452 )≈  1038311.6 , Δ≈-0.00231,infS(m) = 934480.39 , k(m)=  
G(512942454) = 1872511；inf( 512942454 )≈  1869669.2 , Δ≈-0.00175,infS(m) = 934480.40 , k(m)=  
G(512942456) = 1021700；inf( 512942456 )≈  1019433.2 , Δ≈-0.00222,infS(m) = 934480.40 , k(m)=  
G(512942458) = 1182955；inf( 512942458 )≈  1179194.1 , Δ≈-0.00318,infS(m) = 934480.40 , k(m)=  
G(512942460) = 2541272；inf( 512942460 )≈  2535666.1 , Δ≈-0.00221,infS(m) = 934480.41 , k(m)=  
G(512942462) = 1000463；inf( 512942462 )≈  998410.5   , Δ≈-0.00205,infS(m) = 934480.41 , k(m)=
G(512942464) = 935816  ；inf( 512942464 )≈  934480.4   , Δ≈-0.00143,infS(m) = 934480.41 , k(m)= 1
G(512942466) = 1873688；inf( 512942466 )≈  1868960.8 , Δ≈-0.00252,infS(m) = 934480.42 , k(m)= 2
G(512942468) = 992383  ；inf( 512942468 )≈  990212.3   , Δ≈-0.00219,infS(m) = 934480.42 , k(m)=  
G(512942470) = 1292323；inf( 512942470 )≈  1290314.9 , Δ≈-0.00155,infS(m) = 934480.42 , k(m)=
G(512942472) = 2337741；inf( 512942472 )≈  2332754.2 , Δ≈-0.00213,infS(m) = 934480.43 , k(m)=   
G(512942474) = 1040295；inf( 512942474 )≈  1038311.6 , Δ≈-0.00191,infS(m) = 934480.43 , k(m)=
G(512942476) = 936907  ；inf( 512942476 )≈  934480.4   , Δ≈-0.00259,infS(m) = 934480.44 , k(m)= 1
G(512942478) = 1871987；inf( 512942478 )≈  1868960.9 , Δ≈-0.00162,infS(m) = 934480.44 , k(m)= 2
G(512942480) = 1272682；inf( 512942480 )≈  1270404.8 , Δ≈-0.00179,infS(m) = 934480.44 , k(m)=
G(512942482) = 1046622；inf( 512942482 )≈  1043949.5 , Δ≈-0.00255,infS(m) = 934480.45 , k(m)=
G(512942484) = 1874023；inf( 512942484 )≈  1868960.9 , Δ≈-0.00270,infS(m) = 934480.45 , k(m)= 2
G(512942486) = 1140966；inf( 512942486 )≈  1138628.5 , Δ≈-0.00205,infS(m) = 934480.45 , k(m)=
G(512942488) = 970900  ；inf( 512942488 )≈  969090.8   , Δ≈-0.00186,infS(m) = 934480.46 , k(m)=  
G(512942490) = 2589452；inf( 512942490 )≈  2584090   , Δ≈-0.00207,infS(m) = 934480.46 , k(m)=  
G(512942492) = 969182  ；inf( 512942492 )≈  966703.9  , Δ≈-0.00256,infS(m) = 934480.46 , k(m)=  
G(512942494) = 937120  ；inf( 512942494 )≈  934480.5  , Δ≈-0.00282,infS(m) = 934480.47 , k(m)= 1
G(512942496) = 2220483；inf( 512942496 )≈  2215064.8, Δ≈-0.00244,infS(m) = 934480.47 , k(m)=
G(512942498) = 937575  ；inf( 512942498 )≈  935279.9  , Δ≈-0.00245,infS(m) = 934480.48 , k(m)=  
G(512942500) = 1499286；inf( 512942500 )≈  1495168.8, Δ≈-0.00275,infS(m) = 934480.48 , k(m)= 1.6
G(512942502) = 1873102；inf( 512942502 )≈  1868961   , Δ≈-0.00221,infS(m) = 934480.48 , k(m)= 2
G(512942504) = 987663  ；inf( 512942504 )≈  984971.8  , Δ≈-0.00272,infS(m) = 934480.49 , k(m)=
G(512942506) = 995430  ；inf( 512942506 )≈  993047.9  , Δ≈-0.00239,infS(m) = 934480.49 , k(m)=
G(512942508) = 2042841；inf( 512942508 )≈  2038866.5, Δ≈-0.00195,infS(m) = 934480.49 , k(m)=
G(512942510) = 1254049；inf( 512942510 )≈  1251367.8, Δ≈-0.00214,infS(m) = 934480.50 , k(m)=  
G(512942512) = 936506  ；inf( 512942512 )≈  934480.5  , Δ≈-0.00216,infS(m) = 934480.50 , k(m)= 1
G(512942514) = 2246531；inf( 512942514 )≈ 2242753.2 , Δ≈-0.00168,infS(m) = 934480.50 , k(m)= 2.4
G(512942516) = 935956  ；inf( 512942516 )≈  934661.3  , Δ≈-0.00138,infS(m) = 934480.51 , k(m)=
G(512942518) = 1039969；inf( 512942518 )≈  1038311.7, Δ≈-0.00159,infS(m) = 934480.51 , k(m)=  
G(512942520) = 2497917；inf( 512942520 )≈  2491948.1, Δ≈-0.00239,infS(m) = 934480.52 , k(m)=  
G(512942522) = 945553  ；inf( 512942522 )≈  944587.6  , Δ≈-0.00102,infS(m) = 934480.52 , k(m)=  
G(512942524) = 936428  ；inf( 512942524 )≈  934480.5  , Δ≈-0.00208,infS(m) = 934480.52 , k(m)= 1
G(512942526) = 1896181；inf( 512942526 )≈  1890969   , Δ≈-0.00275,infS(m) = 934480.53 , k(m)=  
G(512942528) = 1151134；inf( 512942528 )≈  1148727.3, Δ≈-0.00209,infS(m) = 934480.53 , k(m)=
G(512942530) = 1350407；inf( 512942530 )≈  1347757.8, Δ≈-0.00196,infS(m) = 934480.53 , k(m)=  
time start =13:52:55  ,time end =13:54:32   ,time use =

yangchuanju

重生888@ 发表于 2023-2-17 16:19
我计算后，觉得4123665个素数对，没有毛病！因为：
D（512942430）=2468268
D1=2468268*（7. 11. 13.19 ...

吴的精度0.988，愚公的精度0.99844，还是不如愚公的精度吧！

重生888@

从一种组合最大下限值2468268/4=617067
D（512942432）=617067*1.5=925600
D（512942434）=同上
D（512942436）=617067*3=1851201

D（512942440）=617067*2=1234134
计算又快，精度0.988...

愚工688

若计算值精度达0.999以上时，很容易发生计算值的相对误差为正的情况，那就不是下界计算值了。

例;下面的偶数的素数对的计算，我仅仅验证了5个偶数，其中2个为正相对误差。它们的精度都比较高。

Sp( 520000000 *)≈  1378559 , Δ≈, k(m)= 1.45455
Sp( 520000002 *)≈  1895518.6 , Δ≈, k(m)= 2
Sp( 520000004 *)≈  947759.3 , Δ≈, k(m)= 1
Sp( 520000006 *)≈  947878.2 , Δ≈, k(m)= 1.00013
Sp( 520000008 *)≈  2206423.8 , Δ≈, k(m)= 2.32804
Sp( 520000010 *)≈  1303531.5 , Δ≈, k(m)= 1.37538
Sp( 520000012 *)≈  1223403.1 , Δ≈, k(m)= 1.29084
Sp( 520000014 *)≈  1973643.1 , Δ≈, k(m)= 2.08243
Sp( 520000016 *)≈  947759.3 , Δ≈, k(m)= 1
Sp( 520000018 *)≈  1008656.1 , Δ≈, k(m)= 1.06425
Sp( 520000020 *)≈  2527358.2 , Δ≈, k(m)= 2.66667
Sp( 520000022 *)≈  947759.3 , Δ≈, k(m)= 1
Sp( 520000024 *)≈  975114.9 , Δ≈, k(m)= 1.02886
Sp( 520000026 *)≈  2481406.3 , Δ≈, k(m)= 2.61818
Sp( 520000028 *)≈  947759.4 , Δ≈, k(m)= 1
Sp( 520000030 *)≈  1405388.6 , Δ≈, k(m)= 1.48285
Sp( 520000032 *)≈  1962582.2 , Δ≈, k(m)= 2.07076
Sp( 520000034 *)≈  948106.9 , Δ≈, k(m)= 1.00037
Sp( 520000036 *)≈  947759.4 , Δ≈, k(m)= 1
Sp( 520000038 *)≈  1895518.7 , Δ≈, k(m)= 2
Sp( 520000040 *)≈  1543018.8 , Δ≈, k(m)= 1.62807
Sp( 520000042 *)≈  947759.4 , Δ≈, k(m)= 1
Sp( 520000044 *)≈  1965723.2 , Δ≈, k(m)= 2.07407
Sp( 520000046 *)≈  1024810.8 , Δ≈, k(m)= 1.0813
Sp( 520000048 *)≈  975868.7 , Δ≈, k(m)= 1.02966
G(520000050) = 2527241；Sp( 520000050 *)≈  2527358.4 , Δ≈ 0.00006296, k(m)= 2.66667
G(520000052) = 1147823；Sp( 520000052 *)≈  1148799.3 , Δ≈ 0.00085, k(m)= 1.21212
G(520000054) = 1197285；Sp( 520000054 *)≈  1197199.5 , Δ≈-0.00007099, k(m)= 1.26319
G(520000056) = 2007412；Sp( 520000056 *)≈  2007019.9 , Δ≈-0.0001953, k(m)= 2.11765
G(520000058) = 948264；Sp( 520000058 *)≈  947852.7 , Δ≈-0.000433, k(m)= 1.0001

start time :19:08:49, end time:19:09:52use time :

G(520000000) = 1379055
G(520000002) = 1896963
G(520000004) = 948520
G(520000006) = 947778
G(520000008) = 2205779
G(520000010) = 1304264
G(520000012) = 1224423
G(520000014) = 1974934
G(520000016) = 947911
G(520000018) = 1009455
G(520000020) = 2527114
G(520000022) = 948360
G(520000024) = 975238
G(520000026) = 2482025
G(520000028) = 947050
G(520000030) = 1406679
G(520000032) = 1963419
G(520000034) = 948520
G(520000036) = 947612
G(520000038) = 1894927
G(520000040) = 1542704
G(520000042) = 948741
G(520000044) = 1965982
G(520000046) = 1025543
G(520000048) = 976329

yangchuanju

愚工688 发表于 2023-2-17 19:26
若计算值精度达0.999以上时，很容易发生计算值的相对误差为正的情况，那就不是下界计算值了。

例;下面的 ...

“若计算值精度达0.999以上时，很容易发生计算值的相对误差为正的情况，那就不是下界计算值了。”

精确计算，相对误差有正有负，某段偶数的素数对相对误差的平均值接近于0才好呢！
愚公老师搞的是精确计算，为什么又说是下界（下限）？

吴代业把精确值与下限混在一起，愚公老师不能再混在一起。

yangchuanju

vfbpgyfk
从我的计算方法和套路上讲，精确计算素数对个数公式是：kN/ln(N)^2(k系数是【分类系数】与【动态系数】的乘积)；素数对个数下限计算公式是：N/2ln(N)^2。  发表于 2023-2-17 20:59

回复那宝吉老师：
N/2ln(N)^2不是偶数素数对的下限，该计算值比下包络线值小许多。

重生888@

既然计算即快又准，为何不能连续地多计算一些？为何还要有间断？

年岁大了，就是输入数字倒费劲。上次答应您的51个数，一定完成！

vfbpgyfk

yangchuanju 发表于 2023-2-17 22:12
vfbpgyfk
从我的计算方法和套路上讲，精确计算素数对个数公式是：kN/ln(N)^2(k系数是【分类系数】与【动态 ...

这是通用计算式，着眼点是确保任意偶数的最少素数对都居于此下限之内，主要表现在小偶数范围。你所讲的那种包络线的下限值，应该是阶段性的计算范围。根据实践经验来看，偶数越大，与阶段性计算出来的下包络差距越大。
从较大范围的计算结果上看，素数对的最低点连线是呈逐渐高于那个素数对下限计算曲线的。

愚工688

把55楼的数据按照计算值精度计算，如下：


G(520000000) = 1379055 ；Sp( 520000000 *)≈  1378559    , 精度≈0.99964 , k(m)=
G(520000002) = 1896963 ；Sp( 520000002 *)≈  1895518.6 , 精度≈0.99924 , k(m)= 2
G(520000004) = 948520  ；Sp( 520000004 *)≈  947759.3    , 精度≈0.99920 , k(m)= 1
G(520000006) = 947778  ；Sp( 520000006 *)≈  947878.2    , 精度≈1.00011 , k(m)=
G(520000008) = 2205779 ；Sp( 520000008 *)≈  2206423.8 , 精度≈1.00029 , k(m)=
G(520000010) = 1304264 ；Sp( 520000010 *)≈  1303531.5 , 精度≈0.99944 , k(m)=
G(520000012) = 1224423 ；Sp( 520000012 *)≈  1223403.1 , 精度≈0.99917 , k(m)=  
G(520000014) = 1974934 ；Sp( 520000014 *)≈  1973643.1 , 精度≈0.99935 , k(m)=
G(520000016) = 947911  ；Sp( 520000016 *)≈  947759.3    , 精度≈0.99984 , k(m)= 1
G(520000018) = 1009455 ；Sp( 520000018 *)≈  1008656.1 , 精度≈0.99921 , k(m)=  
G(520000020) = 2527114 ；Sp( 520000020 *)≈  2527358.2 , 精度≈1.00010 , k(m)=  
G(520000022) = 948360  ；Sp( 520000022 *)≈  947759.3    , 精度≈0.99937 , k(m)= 1
G(520000024) = 975238  ；Sp( 520000024 *)≈  975114.9    , 精度≈0.99987 , k(m)=
G(520000026) = 2482025 ；Sp( 520000026 *)≈  2481406.3 , 精度≈0.99975 , k(m)=
G(520000028) = 947050  ；Sp( 520000028 *)≈  947759.4    , 精度≈1.00075 , k(m)= 1
G(520000030) = 1406679 ；Sp( 520000030 *)≈  1405388.6 , 精度≈0.99908 , k(m)=  
G(520000032) = 1963419 ；Sp( 520000032 *)≈  1962582.2 , 精度≈0.99957 , k(m)=  
G(520000034) = 948520  ；Sp( 520000034 *)≈  948106.9    , 精度≈0.99956 , k(m)=  
G(520000036) = 947612  ；Sp( 520000036 *)≈  947759.4    , 精度≈1.00016 , k(m)= 1
G(520000038) = 1894927 ；Sp( 520000038 *)≈  1895518.7 , 精度≈1.00031 , k(m)= 2
G(520000040) = 1542704 ；Sp( 520000040 *)≈  1543018.8 , 精度≈1.00020 , k(m)=  
G(520000042) = 948741  ；Sp( 520000042 *)≈  947759.4    , 精度≈0.99896 , k(m)= 1
G(520000044) = 1965982 ；Sp( 520000044 *)≈  1965723.2 , 精度≈0.99987 , k(m)=
G(520000046) = 1025543 ；Sp( 520000046 *)≈  1024810.8 , 精度≈0.99929 , k(m)=
G(520000048) = 976329  ；Sp( 520000048 *)≈  975868.7    , 精度≈0.99953 , k(m)=
G(520000050) = 2527241 ；Sp( 520000050 *)≈  2527358.4 , 精度≈1.00006 , k(m)=
G(520000052) = 1147823 ；Sp( 520000052 *)≈  1148799.3 , 精度≈1.00085 , k(m)=  
G(520000054) = 1197285 ；Sp( 520000054 *)≈  1197199.5 , 精度≈0.99993 , k(m)=
G(520000056) = 2007412 ；Sp( 520000056 *)≈  2007019.9 , 精度≈0.99980 , k(m)=
G(520000058) = 948264  ；Sp( 520000058 *)≈  947852.7    , 精度≈0.99957 ,  k(m)=

start time :19:08:49, end time:19:09:52use time :