高精度计算大偶数表为两个素数和的表法数值的实例（以当天日期为随机数选择偶数）

愚工688

用Sp( m *)=Sp( m )/(1+μ) 来计算350亿-550亿区间偶数的素对数量，这里的μ=0.156494,

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(35000000000) = 68412556, Sp(35000000000 *) ≈  68356579.9  ,Δ≈-0.000818214 ,
G(35000000002) = 48894586, Sp(35000000002 *) ≈  48850013.3  ,Δ≈-0.000911608 ,
G(35000000004) = 85531578, Sp(35000000004 *) ≈  85455556.4  ,Δ≈-0.000888813 ,
G(35000000006) = 42755368, Sp(35000000006 *) ≈  42723589.9  ,Δ≈-0.000743254 ,

2000亿的计算:
Sp(250000000000 *) =  346137314.3  Δ= 0.000009393   k(m)= 1.33333
Sp(250000000002 *) =  519517806.9  Δ= 0.000008266   k(m)= 2.0012
Sp(250000000004 *) =  259622689.9  Δ= 0.000134205   k(m)= 1.00008
Sp(250000000006 *) =  259602985.7  Δ= 0.000003843   k(m)= 1


G(250000000000) = 346134063
G(250000000002) = 519474867
G(250000000004) = 259587852
G(250000000006) = 259601988
G(250000000008) = 576847692

用Sp(m *)=Sp(m)/(1+μ) 来计算9000亿-13000亿的大偶数的素对数量，这里的μ=0.16842,
G(1200000000000)=2944418095；Sp(1200000000000 *) =  2946781907.9  Δ=0.0008028    k(m)= 2.66667
G(1200000000002)=1125773371；Sp(1200000000002 *) =  1126710729.5  Δ=0.0008326    k(m)= 1.01961
G(1200000000004)=1204534461；Sp(1200000000004 *) =  1205501689.6  Δ=0.0008030    k(m)= 1.09091
G(1200000000006)=2208342854；Sp(1200000000006 *) =  2210086430.9  Δ=0.0007895    k(m)= 2

愚工688

以今天日期的万倍为随机偶数的素对下界计算，12个偶数的连乘式的计算使用了22分钟多，以此推算，如果计算日期十万倍的12个偶数，至少需要240分钟以上。而素对数量的变化规律不会有变化，所以没有必要计算太大的偶数。相对误差今天不计算了，数据直接贴上来。

inf( 202102160000 )≈  321420325.2 , Δ≈,infS(m) = 213052653 , k(m)= 1.50864
inf( 202102160002 )≈  213058971.5 , Δ≈,infS(m) = 213052653 , k(m)= 1.00003
inf( 202102160004 )≈  433327429.8 , Δ≈,infS(m) = 213052653 , k(m)= 2.0339
inf( 202102160006 )≈  217230156 , Δ≈,infS(m) = 213052653 , k(m)= 1.01961
inf( 202102160008 )≈  255663183.6 , Δ≈,infS(m) = 213052653 , k(m)= 1.2
inf( 202102160010 )≈  609372905.2 , Δ≈,infS(m) = 213052653.01 , k(m)= 2.8602
inf( 202102160012 )≈  213052653 , Δ≈,infS(m) = 213052653.01 , k(m)= 1
inf( 202102160014 )≈  213056998.3 , Δ≈,infS(m) = 213052653.01 , k(m)= 1.00002
inf( 202102160016 )≈  443364866.9 , Δ≈,infS(m) = 213052653.01 , k(m)= 2.08101
inf( 202102160018 )≈  253061808.1 , Δ≈,infS(m) = 213052653.02 , k(m)= 1.18779
inf( 202102160020 )≈  288071192.8 , Δ≈,infS(m) = 213052653.02 , k(m)= 1.35211
inf( 202102160022 )≈  513187597 , Δ≈,infS(m) = 213052653.02 , k(m)= 2.40874
time start =20:02:48  ,time end =20:25:07   ,time use =

真值的筛选程序采用黄博士赠予的软件，速度快了一些，但太大偶数也会溢出计算范围：
G(202102160000) = 322016532
G(202102160002) = 213450910
G(202102160004) = 434115298
G(202102160006) = 217622993
G(202102160008) = 256120362
G(202102160010) = 610478254
G(202102160012) = 213428156
G(202102160014) = 213449516
G(202102160016) = 444179041
G(202102160018) = 253524361
G(202102160020) = 288582702
G(202102160022) = 514128982

count = 12, algorithm = 2, working threads = 2, time use 105.936 sec

愚工688

愚工688 发表于 2021-2-16 12:46
以今天日期的万倍为随机偶数的素对下界计算，12个偶数的连乘式的计算使用了22分钟多，以此推算，如果计算日 ...

622#偶数素对下界计算值的相对误差数据：


        G(202102160000) = 322016532；
inf( 202102160000 )≈  321420325.2 , Δ≈-0.00185 ,infS(m) = 213052653 , k(m)= 1.50864
        G(202102160002) = 213450910；
inf( 202102160002 )≈  213058971.5 , Δ≈-0.00184 ,infS(m) = 213052653 , k(m)= 1.00003
        G(202102160004) = 434115298；
inf( 202102160004 )≈  433327429.8 , Δ≈-0.00181 ,infS(m) = 213052653 , k(m)= 2.0339
        G(202102160006) = 217622993；
inf( 202102160006 )≈  217230156.0 , Δ≈-0.00181 ,infS(m) = 213052653 , k(m)= 1.01961
        G(202102160008) = 256120362；
inf( 202102160008 )≈  255663183.6 , Δ≈-0.00179 ,infS(m) = 213052653 , k(m)= 1.2
        G(202102160010) = 610478254；
inf( 202102160010 )≈  609372905.2 , Δ≈-0.00181 ,infS(m) = 213052653.01 , k(m)= 2.8602
        G(202102160012) = 213428156；
inf( 202102160012 )≈  213052653.0 , Δ≈-0.00176 ,infS(m) = 213052653.01 , k(m)= 1
        G(202102160014) = 213449516；
inf( 202102160014 )≈  213056998.3 , Δ≈-0.00184 ,infS(m) = 213052653.01 , k(m)= 1.00002
        G(202102160016) = 444179041；
inf( 202102160016 )≈  443364866.9 , Δ≈-0.00183 ,infS(m) = 213052653.01 , k(m)= 2.08101
        G(202102160018) = 253524361；
inf( 202102160018 )≈  253061808.1 , Δ≈-0.00182 ,infS(m) = 213052653.02 , k(m)= 1.18779
        G(202102160020) = 288582702；
inf( 202102160020 )≈  288071192.8 , Δ≈-0.00177 ,infS(m) = 213052653.02 , k(m)= 1.35211
        G(202102160022) = 514128982；
inf( 202102160022 )≈  513187597.0 , Δ≈-0.00183 ,infS(m) = 213052653.02 , k(m)= 2.40874
time start =20:02:48  ,time end =20:25:07   ,time use =

其中可以看到：
1，所有偶数的下界计算值的相对误差都很小，并且比较接近；
2，它们的,infS(m)  呈现近乎线性地增大，基本每4-5个偶数增大0.01 ；
即偶数大10000，,infS(m) 会增大11左右，
偶数大1000000，,infS(m)值 会增大1100左右；因此大偶数的区域素对下界值会越来越大。
3，区域素对下界值,infS(m)与偶数的素对下界计算值,inf(m) 的关系：infS(m)=inf(m)/ k(m)；
4，不大范围内，偶数的素对数量的变化主要是由偶数含有的奇素因子的综合作用所影响的；

总的来说，偶数M能够形成的素数对的数量是有规律的，也是能够比较精确的计算出来的。

例：
inf( 202103160000 )≈  681774696.7 , Δ≈,infS(m) = 213053707.18 , k(m)= 3.20001
inf( 202103160002 )≈  213053707.2 , Δ≈,infS(m) = 213053707.18 , k(m)= 1
inf( 202103160004 )≈  215380543.9 , Δ≈,infS(m) = 213053707.18 , k(m)= 1.01092
time start =14:29:09  ,time end =14:34:33   ,time use =
实际验证大1000000的偶数的区域下界值增大了1054，略小于1100。

愚工688

今天是2021年3月15日。 以今天日期的千倍开始的连续偶数的素对数量的计算，看看计算值的相对误差怎么样？

G(20210315000) = 34533479；
inf( 20210315000 )≈  34514743.1 , Δ≈-0.0005425,infS(m) = 25886057.31 , k(m)= 1.33333
   G(20210315002) = 26172802；
inf( 20210315002 )≈  26156924.4 , Δ≈-0.0006067,infS(m) = 25886057.31 , k(m)= 1.01046
   G(20210315004) = 61177399；
inf( 20210315004 )≈  61139182.4 , Δ≈-0.0006247,infS(m) = 25886057.31 , k(m)= 2.36186
   G(20210315006) = 32678448；
inf( 20210315006 )≈  32660722.6 , Δ≈-0.0005424,infS(m) = 25886057.32 , k(m)= 1.26171
   G(20210315008) = 28749945；
inf( 20210315008 )≈  28735041.4 , Δ≈-0.0005184,infS(m) = 25886057.32 , k(m)= 1.11006
   G(20210315010) = 75224659；
inf( 20210315010 )≈  75178331.2 , Δ≈-0.0006159,infS(m) = 25886057.32 , k(m)= 2.9042
   G(20210315012) = 25963202；
inf( 20210315012 )≈  25948519.4 , Δ≈-0.0005655,infS(m) = 25886057.32 , k(m)= 1.00241
   G(20210315014) = 25898266；
inf( 20210315014 )≈  25886469.4 , Δ≈-0.0004555,infS(m) = 25886057.33 , k(m)= 1.00002
   G(20210315016) = 51874564；
inf( 20210315016 )≈  51843524.5 , Δ≈-0.0005984,infS(m) = 25886057.33 , k(m)= 2.00276
   G(20210315018) = 26526591；
inf( 20210315018 )≈  26517424.6 , Δ≈-0.0003456,infS(m) = 25886057.33 , k(m)= 1.02439
   G(20210315020) = 44292566；
inf( 20210315020 )≈  44260556.1 , Δ≈-0.0007227,infS(m) = 25886057.33 , k(m)= 1.70982
   G(20210315022) = 51804510；
inf( 20210315022 )≈  51772114.7 , Δ≈-0.0006253,infS(m) = 25886057.34 , k(m)= 2
time start =16:36:14  ,time end =16:40:30   ,time use =

愚工688

今天是2021年3月15日。 以今天日期的千倍开始的连续偶数的素对数量的计算，看看计算值的相对误差怎么样？

G(20210315000) = 34533479；
inf( 20210315000 )≈  34514743.1 , Δ≈-0.0005425,infS(m) = 25886057.31 , k(m)= 1.33333
   G(20210315002) = 26172802；
inf( 20210315002 )≈  26156924.4 , Δ≈-0.0006067,infS(m) = 25886057.31 , k(m)= 1.01046
   G(20210315004) = 61177399；
inf( 20210315004 )≈  61139182.4 , Δ≈-0.0006247,infS(m) = 25886057.31 , k(m)= 2.36186
   G(20210315006) = 32678448；
inf( 20210315006 )≈  32660722.6 , Δ≈-0.0005424,infS(m) = 25886057.32 , k(m)= 1.26171
   G(20210315008) = 28749945；
inf( 20210315008 )≈  28735041.4 , Δ≈-0.0005184,infS(m) = 25886057.32 , k(m)= 1.11006
   G(20210315010) = 75224659；
inf( 20210315010 )≈  75178331.2 , Δ≈-0.0006159,infS(m) = 25886057.32 , k(m)= 2.9042
   G(20210315012) = 25963202；
inf( 20210315012 )≈  25948519.4 , Δ≈-0.0005655,infS(m) = 25886057.32 , k(m)= 1.00241
   G(20210315014) = 25898266；
inf( 20210315014 )≈  25886469.4 , Δ≈-0.0004555,infS(m) = 25886057.33 , k(m)= 1.00002
   G(20210315016) = 51874564；
inf( 20210315016 )≈  51843524.5 , Δ≈-0.0005984,infS(m) = 25886057.33 , k(m)= 2.00276
   G(20210315018) = 26526591；
inf( 20210315018 )≈  26517424.6 , Δ≈-0.0003456,infS(m) = 25886057.33 , k(m)= 1.02439
   G(20210315020) = 44292566；
inf( 20210315020 )≈  44260556.1 , Δ≈-0.0007227,infS(m) = 25886057.33 , k(m)= 1.70982
   G(20210315022) = 51804510；
inf( 20210315022 )≈  51772114.7 , Δ≈-0.0006253,infS(m) = 25886057.34 , k(m)= 2
time start =16:36:14  ,time end =16:40:30   ,time use =

njzz_yy

愚工688 先生：能不能提供以2021年的10倍，百倍，千倍，万倍，偶数的哥德巴赫素数组个数？比您提供日期的对应个数，小约万倍，谢谢！

愚工688

njzz_yy 发表于 2021-4-1 12:43
愚工688 先生：能不能提供以2021年的10倍，百倍，千倍，万倍，偶数的哥德巴赫素数组个数？比您提供日期的对 ...

  Xi(M)=t2*c1*M/(logM)^2   

  G( 20210 ) = 239              ;Xi(M)≈ 224.94          δxi(M)≈?-0.05900
  G( 202100 ) = 1486            ;Xi(M)≈ 1456.39         δxi(M)≈?-0.01992
  G( 2021000 ) = 10255          ;Xi(M)≈ 10155.85        δxi(M)≈?-0.00967
  G( 20210000 ) = 74618         ;Xi(M)≈ 74603.43        δxi(M)≈?-0.00020
  G( 202100000 ) = 569900       ;Xi(M)≈ 569610.8        δxi(M)≈?-0.00051
  G( 2021000000 ) = 4479670     ;Xi(M)≈ 4480777.89      δxi(M)≈? 0.000247
  G( 20210000000 ) = 36154174   ;Xi(M)≈ 36094443.16     δxi(M)≈?-0.001652
  G( 202100000000 ) = 298007468

愚工688

愚工688 发表于 2021-4-2 03:28
Xi(M)=t2*c1*M/(logM)^2   

  G( 20210 ) = 239              ;Xi(M)≈ 224.94          δxi(M)≈? ...

20210:50:2

G(20210) = 239
G(20212) = 179
G(20214) = 339
G(20216) = 218
G(20218) = 192
G(20220) = 455
G(20222) = 169
G(20224) = 177
G(20226) = 337
G(20228) = 175
G(20230) = 305
G(20232) = 340
G(20234) = 170
G(20236) = 167
G(20238) = 333
G(20240) = 259
G(20242) = 177
G(20244) = 412
G(20246) = 181
G(20248) = 165
G(20250) = 466
G(20252) = 171
G(20254) = 201
G(20256) = 338
G(20258) = 208
G(20260) = 221
G(20262) = 375
G(20264) = 187
G(20266) = 162
G(20268) = 339
G(20270) = 226
G(20272) = 200
G(20274) = 364
G(20276) = 176
G(20278) = 177
G(20280) = 497
G(20282) = 163
G(20284) = 181
G(20286) = 427
G(20288) = 177
G(20290) = 224
G(20292) = 365
G(20294) = 169
G(20296) = 169
G(20298) = 351
G(20300) = 283
G(20302) = 174
G(20304) = 342
G(20306) = 212
G(20308) = 166

202100:50:2

G(202100) = 1486
G(202102) = 1096
G(202104) = 2600
G(202106) = 1088
G(202108) = 1049
G(202110) = 2892
G(202112) = 1125
G(202114) = 1198
G(202116) = 2138
G(202118) = 1290
G(202120) = 1486
G(202122) = 2270
G(202124) = 1248
G(202126) = 1064
G(202128) = 2153
G(202130) = 1591
G(202132) = 1314
G(202134) = 2197
G(202136) = 1187
G(202138) = 1056
G(202140) = 2860
G(202142) = 1090
G(202144) = 1080
G(202146) = 2562
G(202148) = 1064
G(202150) = 1572
G(202152) = 2116
G(202154) = 1083
G(202156) = 1080
G(202158) = 2362
G(202160) = 1830
G(202162) = 1107
G(202164) = 2291
G(202166) = 1112
G(202168) = 1108
G(202170) = 3033
G(202172) = 1035
G(202174) = 1291
G(202176) = 2325
G(202178) = 1058
G(202180) = 1596
G(202182) = 2204
G(202184) = 1095
G(202186) = 1113
G(202188) = 2688
G(202190) = 1435
G(202192) = 1076
G(202194) = 2216
G(202196) = 1127
G(202198) = 1225

2021000:50:2

G(2021000) = 10255
G(2021002) = 7330
G(2021004) = 14838
G(2021006) = 8094
G(2021008) = 8208
G(2021010) = 21564
G(2021012) = 8864
G(2021014) = 7757
G(2021016) = 14858
G(2021018) = 7343
G(2021020) = 9805
G(2021022) = 14700
G(2021024) = 7449
G(2021026) = 8931
G(2021028) = 15745
G(2021030) = 11569
G(2021032) = 8012
G(2021034) = 14678
G(2021036) = 7416
G(2021038) = 7363
G(2021040) = 23609
G(2021042) = 7335
G(2021044) = 7382
G(2021046) = 14818
G(2021048) = 7403
G(2021050) = 10006
G(2021052) = 16762
G(2021054) = 8973
G(2021056) = 7769
G(2021058) = 16105
G(2021060) = 9944
G(2021062) = 7846
G(2021064) = 14765
G(2021066) = 7475
G(2021068) = 9767
G(2021070) = 19638
G(2021072) = 7270
G(2021074) = 8165
G(2021076) = 15198
G(2021078) = 7506
G(2021080) = 9850
G(2021082) = 17630
G(2021084) = 8088
G(2021086) = 7666
G(2021088) = 15163
G(2021090) = 9844
G(2021092) = 7396
G(2021094) = 15056
G(2021096) = 10644


20210000:50:2

G(20210000) = 74618
G(20210002) = 54657
G(20210004) = 107326
G(20210006) = 53949
G(20210008) = 79717
G(20210010) = 142883
G(20210012) = 53697
G(20210014) = 60523
G(20210016) = 110982
G(20210018) = 54606
G(20210020) = 71771
G(20210022) = 128652
G(20210024) = 53663
G(20210026) = 53602
G(20210028) = 107295
G(20210030) = 71484
G(20210032) = 54515
G(20210034) = 123909
G(20210036) = 71501
G(20210038) = 53747
G(20210040) = 143040
G(20210042) = 60091
G(20210044) = 54673
G(20210046) = 107481
G(20210048) = 55000
G(20210050) = 87706
G(20210052) = 107057
G(20210054) = 56268
G(20210056) = 53478
G(20210058) = 119097
G(20210060) = 77835
G(20210062) = 53940
G(20210064) = 128709
G(20210066) = 56217
G(20210068) = 53843
G(20210070) = 142872
G(20210072) = 56676
G(20210074) = 53570
G(20210076) = 114347
G(20210078) = 66259
G(20210080) = 79352
G(20210082) = 107870
G(20210084) = 53728
G(20210086) = 60165
G(20210088) = 107266
G(20210090) = 71439
G(20210092) = 64638
G(20210094) = 109661
G(20210096) = 55607
G(20210098) = 53876

202100000:50:2

G(202100000) = 569900
G(202100002) = 445387
G(202100004) = 815689
G(202100006) = 416094
G(202100008) = 458460
G(202100010) = 1306277
G(202100012) = 440385
G(202100014) = 424369
G(202100016) = 830423
G(202100018) = 408371
G(202100020) = 554763
G(202100022) = 820651
G(202100024) = 503446
G(202100026) = 408667
G(202100028) = 889551
G(202100030) = 603398
G(202100032) = 408653
G(202100034) = 855590
G(202100036) = 446910
G(202100038) = 489501
G(202100040) = 1088836
G(202100042) = 408277
G(202100044) = 409612
G(202100046) = 872143
G(202100048) = 411489
G(202100050) = 543980
G(202100052) = 1087475
G(202100054) = 444951
G(202100056) = 408763
G(202100058) = 815982
G(202100060) = 546889
G(202100062) = 414725
G(202100064) = 816510
G(202100066) = 489649
G(202100068) = 420940
G(202100070) = 1117141
G(202100072) = 423444
G(202100074) = 480228
G(202100076) = 829349
G(202100078) = 408145
G(202100080) = 806888
G(202100082) = 824104
G(202100084) = 408897
G(202100086) = 418005
G(202100088) = 815234
G(202100090) = 545115
G(202100092) = 408210
G(202100094) = 1001213
G(202100096) = 453576
G(202100098) = 422556

愚工688

今天是2021-04-03日，以今天日期的千倍为起始点的连续12个偶数的素对下界计算值的计算精度：

G(20210403000) = 69770347；
inf( 20210403000 )≈  69728102.7 , jd ≈0.999395,infS(m) = 25886170.02 , k(m)= 2.69364
G(20210403002) = 28214531；
inf( 20210403002 )≈  28200101.6 , jd ≈0.999489,infS(m) = 25886170.02 , k(m)= 1.08939
G(20210403004) = 29776813；
inf( 20210403004 )≈  29754218.4 , jd ≈0.999241,infS(m) = 25886170.03 , k(m)= 1.14943
G(20210403006) = 51941637；
inf( 20210403006 )≈  51914182.1 , jd ≈0.999471,infS(m) = 25886170.03 , k(m)= 2.00548
G(20210403008) = 25907685；
inf( 20210403008 )≈  25892428.0 , jd ≈0.999411,infS(m) = 25886170.03 , k(m)= 1.00024
G(20210403010) = 41477486；
inf( 20210403010 )≈  41449707.5 , jd ≈0.999330,infS(m) = 25886170.03 , k(m)= 1.60123
G(20210403012) = 55758283；
inf( 20210403012 )≈  55721341.4 , jd ≈0.999337,infS(m) = 25886170.04 , k(m)= 2.15255
G(20210403014) = 25952234；
inf( 20210403014 )≈  25935855.6 , jd ≈0.999369,infS(m) = 25886170.04 , k(m)= 1.00192
G(20210403016) = 25902622；
inf( 20210403016 )≈  25887831.4 , jd ≈0.999429,infS(m) = 25886170.04 , k(m)= 1.00006
G(20210403018) = 59838359；
inf( 20210403018 )≈  59801205.7 , jd ≈0.999379,infS(m) = 25886170.04 , k(m)= 2.31016
G(20210403020) = 34934727；
inf( 20210403020 )≈  34911616.3 , jd ≈0.999338,infS(m) = 25886170.05 , k(m)= 1.34866
G(20210403022) = 25953068；
inf( 20210403022 )≈  25938002.3 , jd ≈0.999419,infS(m) = 25886170.05 , k(m)= 1.002
time start =10:57:27  ,time end =11:01:36   ,time use =

愚工688

愚工688 发表于 2021-4-3 02:52
20210:50:2

G(20210) = 239

回复：njzz-yy 的谢谢愚工688 ！  发表于 2021-4-3 08:01

只要不是太大的偶数（1*10^14及以上）还是容易的。