n以内所有质数的乘积
我们知道，可以用斯特林公式来估计n以内所有正整数的乘积(n!)。我的问题是，能否用一个简单的方法求出n以内所有质数的乘积的近似值。如n=19，求出 2*3*5*7*11*13*17*19的近似值 当$p$足够大时，$p# = e^p$（此处的$p$，最好是个素数）。 谢谢楼上。不过，我的测试数据表明，误差还是不小的，不知道有没有更精确的公式。
product(2)=2 e^2=7.38906 ratio=3.69453
product(3)=6 e^3=20.0855 ratio=3.34759
product(5)=30 e^5=148.413 ratio=4.94711
product(7)=210 e^7=1096.63 ratio=5.22206
product(11)=2310 e^11=59874.1 ratio=25.9195
product(13)=30030 e^13=442413 ratio=14.7324
product(17)=510510 e^17=2.4155e+007 ratio=47.3153
product(19)=9.69969e+006 e^19=1.78482e+008 ratio=18.4008
product(23)=2.23093e+008 e^23=9.7448e+009 ratio=43.6805
product(29)=6.46969e+009 e^29=3.93133e+012 ratio=607.654
product(31)=2.0056e+011e^31=2.90488e+013 ratio=144.838
product(37)=7.42074e+012 e^37=1.17191e+016 ratio=1579.24
product(41)=3.0425e+014e^41=6.39843e+017 ratio=2103.02
product(43)=1.30828e+016 e^43=4.72784e+018 ratio=361.379
product(47)=6.1489e+017e^47=2.58131e+020 ratio=419.801
product(53)=3.25892e+019 e^53=1.04138e+023 ratio=3195.47
product(59)=1.92276e+021 e^59=4.20121e+025 ratio=21849.9
product(61)=1.17288e+023 e^61=3.1043e+026 ratio=2646.72
product(67)=7.85832e+024 e^67=1.25236e+029 ratio=15936.8
product(71)=5.57941e+026 e^71=6.83767e+030 ratio=12255.2
product(73)=4.07297e+028 e^73=5.05239e+031 ratio=1240.47
product(79)=3.21764e+030 e^79=2.03828e+034 ratio=6334.7
product(83)=2.67065e+032 e^83=1.11286e+036 ratio=4167.02
product(89)=2.37687e+034 e^89=4.48961e+038 ratio=18888.7
product(97)=2.30557e+036 e^97=1.33833e+042 ratio=580479
product(101)=2.32862e+038 e^101=7.30706e+043 ratio=313793
product(103)=2.39848e+040 e^103=5.39923e+044 ratio=22511
product(107)=2.56638e+042 e^107=2.94788e+046 ratio=11486.5
product(109)=2.79735e+044 e^109=2.1782e+047 ratio=778.667
product(113)=3.16101e+046 e^113=1.18926e+049 ratio=376.228
product(127)=4.01448e+048 e^127=1.43021e+055 ratio=3.56263e+006
product(131)=5.25896e+050 e^131=7.80867e+056 ratio=1.48483e+006
product(137)=7.20478e+052 e^137=3.15024e+059 ratio=4.37243e+006
product(139)=1.00146e+055 e^139=2.32773e+060 ratio=232433
product(149)=1.49218e+057 e^149=5.12717e+064 ratio=3.43602e+007
product(151)=2.2532e+059 e^151=3.7885e+065 ratio=1.68139e+006
product(157)=3.53752e+061 e^157=1.52839e+068 ratio=4.32051e+006
product(163)=5.76615e+063 e^163=6.16596e+070 ratio=1.06934e+007
product(167)=9.62947e+065 e^167=3.3665e+072 ratio=3.49604e+006
product(173)=1.6659e+068 e^173=1.35814e+075 ratio=8.15261e+006
product(179)=2.98196e+070 e^179=5.47914e+077 ratio=1.83743e+007
product(181)=5.39735e+072 e^181=4.04857e+078 ratio=750103
product(191)=1.03089e+075 e^191=8.91756e+082 ratio=8.65032e+007
product(193)=1.98962e+077 e^193=6.58924e+083 ratio=3.3118e+006
product(197)=3.91956e+079 e^197=3.5976e+085 ratio=917859
product(199)=7.79992e+081 e^199=2.65829e+086 ratio=34080.9
product(211)=1.64578e+084 e^211=4.32649e+091 ratio=2.62883e+007
product(223)=3.6701e+086 e^223=7.04157e+096 ratio=1.91863e+010
product(227)=8.33112e+088 e^227=3.84457e+098 ratio=4.61471e+009
product(229)=1.90783e+091 e^229=2.84077e+099 ratio=1.48901e+008
product(233)=4.44524e+093 e^233=1.55101e+101 ratio=3.48915e+007
product(239)=1.06241e+096 e^239=6.25722e+103 ratio=5.88964e+007
product(241)=2.56041e+098 e^241=4.62349e+104 ratio=1.80576e+006
product(251)=6.42663e+100 e^251=1.01839e+109 ratio=1.58464e+008
product(257)=1.65164e+103 e^257=4.10849e+111 ratio=2.48751e+008
product(263)=4.34383e+105 e^263=1.65748e+114 ratio=3.81572e+008
product(269)=1.16849e+108 e^269=6.68676e+116 ratio=5.72257e+008
product(271)=3.16661e+110 e^271=4.94088e+117 ratio=1.56031e+007
product(277)=8.7715e+112 e^277=1.99329e+120 ratio=2.27247e+007
product(281)=2.46479e+115 e^281=1.0883e+122 ratio=4.41539e+006
product(283)=6.97536e+117 e^283=8.04152e+122 ratio=115285
product(293)=2.04378e+120 e^293=1.77126e+127 ratio=8.66661e+006
product(307)=6.2744e+122 e^307=2.13013e+133 ratio=3.39495e+010
product(311)=1.95134e+125 e^311=1.16301e+135 ratio=5.96007e+009
product(313)=6.10769e+127 e^313=8.59355e+135 ratio=1.407e+008
product(317)=1.93614e+130 e^317=4.69192e+137 ratio=2.42334e+007
product(331)=6.40862e+132 e^331=5.64253e+143 ratio=8.80459e+010
product(337)=2.1597e+135 e^337=2.27636e+146 ratio=1.05401e+011
product(347)=7.49417e+137 e^347=5.01401e+150 ratio=6.69054e+012
product(349)=2.61547e+140 e^349=3.70488e+151 ratio=1.41653e+011
product(353)=9.2326e+142 e^353=2.0228e+153 ratio=2.19093e+010
product(359)=3.3145e+145 e^359=8.16054e+155 ratio=2.46207e+010
product(367)=1.21642e+148 e^367=2.43262e+159 ratio=1.99982e+011
product(373)=4.53726e+150 e^373=9.8139e+161 ratio=2.16296e+011
product(379)=1.71962e+153 e^379=3.95921e+164 ratio=2.30238e+011
product(383)=6.58615e+155 e^383=2.16166e+166 ratio=3.28213e+010
product(389)=2.56201e+158 e^389=8.72074e+168 ratio=3.40387e+010
product(397)=1.01712e+161 e^397=2.59962e+172 ratio=2.55587e+011
product(401)=4.07864e+163 e^401=1.41934e+174 ratio=3.47994e+010
product(409)=1.66817e+166 e^409=4.231e+177 ratio=2.53632e+011
product(419)=6.98961e+168 e^419=9.3194e+181 ratio=1.33332e+013
product(421)=2.94263e+171 e^421=6.88616e+182 ratio=2.34014e+011
product(431)=1.26827e+174 e^431=1.51678e+187 ratio=1.19594e+013
product(433)=5.49162e+176 e^433=1.12075e+188 ratio=2.04085e+011
product(439)=2.41082e+179 e^439=4.52145e+190 ratio=1.87548e+011
product(443)=1.06799e+182 e^443=2.46863e+192 ratio=2.31146e+010
product(449)=4.79529e+184 e^449=9.95915e+194 ratio=2.07686e+010
product(457)=2.19145e+187 e^457=2.96878e+198 ratio=1.35471e+011
product(461)=1.01026e+190 e^461=1.6209e+200 ratio=1.60444e+010
product(463)=4.67749e+192 e^463=1.19769e+201 ratio=2.56054e+008
product(467)=2.18439e+195 e^467=6.53918e+202 ratio=2.9936e+007
product(479)=1.04632e+198 e^479=1.06428e+208 ratio=1.01716e+010
product(487)=5.09559e+200 e^487=3.17258e+211 ratio=6.22613e+010
product(491)=2.50193e+203 e^491=1.73217e+213 ratio=6.92332e+009
product(499)=1.24847e+206 e^499=5.16353e+216 ratio=4.1359e+010
product(503)=6.27978e+208 e^503=2.81919e+218 ratio=4.48931e+009
product(509)=3.19641e+211 e^509=1.13734e+221 ratio=3.55819e+009
product(521)=1.66533e+214 e^521=1.85108e+226 ratio=1.11154e+012
product(523)=8.70967e+216 e^523=1.36777e+227 ratio=1.57041e+010
product(541)=4.71193e+219 e^541=8.98079e+234 ratio=1.90597e+015
product(547)=2.57743e+222 e^547=3.62311e+237 ratio=1.40571e+015
product(557)=1.43563e+225 e^557=7.98043e+241 ratio=5.55885e+016
product(563)=8.08258e+227 e^563=3.21954e+244 ratio=3.9833e+016
product(569)=4.59899e+230 e^569=1.29885e+247 ratio=2.82422e+016
product(571)=2.62602e+233 e^571=9.5973e+247 ratio=3.65469e+014
product(577)=1.51521e+236 e^577=3.87183e+250 ratio=2.5553e+014
product(587)=8.89431e+238 e^587=8.52827e+254 ratio=9.58846e+015
product(593)=5.27432e+241 e^593=3.44055e+257 ratio=6.5232e+015
product(599)=3.15932e+244 e^599=1.38802e+260 ratio=4.3934e+015
product(601)=1.89875e+247 e^601=1.02561e+261 ratio=5.40151e+013
product(607)=1.15254e+250 e^607=4.13762e+263 ratio=3.58999e+013
product(613)=7.06508e+252 e^613=1.66923e+266 ratio=2.36265e+013
product(617)=4.35916e+255 e^617=9.11371e+267 ratio=2.09071e+012
product(619)=2.69832e+258 e^619=6.73417e+268 ratio=2.49569e+010
product(631)=1.70264e+261 e^631=1.09602e+274 ratio=6.43718e+012
product(641)=1.09139e+264 e^641=2.41414e+278 ratio=2.21199e+014
product(643)=7.01765e+266 e^643=1.78382e+279 ratio=2.54191e+012
product(647)=4.54042e+269 e^647=9.73935e+280 ratio=2.14503e+011
product(653)=2.96489e+272 e^653=3.92913e+283 ratio=1.32522e+011
product(659)=1.95386e+275 e^659=1.58513e+286 ratio=8.11277e+010
product(661)=1.2915e+278 e^661=1.17126e+287 ratio=9.06894e+008
product(673)=8.69182e+280 e^673=1.90628e+292 ratio=2.19319e+011
product(677)=5.88436e+283 e^677=1.04079e+294 ratio=1.76874e+010
product(683)=4.01902e+286 e^683=4.19886e+296 ratio=1.04475e+010
product(691)=2.77714e+289 e^691=1.25166e+300 ratio=4.50701e+010
$\lim_{n->\infty}(p_n#)^(1//p_n)=e$，其中 $p_n$ 表示第 $n$ 个正素数。
虽然该公式当 $n$ 比较小时，误差比较大，但可以反映最终的逼近趋势。 3# liangbch
4# gxqcn
:lol
http://mathworld.wolfram.com/PrimeProducts.html try
${n!*ln^n(n)}/{exp(n/{ln(n)})}$ 我这个是前n个素数乘积的估计，需要改变一下 数据显示2#的结果偏大，而7楼的结果偏小。不过7楼的误差更小一些。下面是测试数据。product(2)=2 p2=0 ratio=0
product(3)=6 p2=0.0536503 ratio=0.00894171
product(5)=30 p2=0.5185 ratio=0.0172833
product(7)=210 p2=4.94908 ratio=0.023567
product(11)=2310 p2=57.9883 ratio=0.0251032
product(13)=30030 p2=836.96 ratio=0.0278708
product(17)=510510 p2=14588.4 ratio=0.0285761
product(19)=9.69969e+006 p2=300803 ratio=0.0310116
product(23)=2.23093e+008 p2=7.20659e+006 ratio=0.0323031
product(29)=6.46969e+009 p2=1.97607e+008 ratio=0.0305435
product(31)=2.0056e+011p2=6.12385e+009 ratio=0.0305337
product(37)=7.42074e+012 p2=2.12218e+011 ratio=0.028598
product(41)=3.0425e+014p2=8.15002e+012 ratio=0.0267872
product(43)=1.30828e+016 p2=3.44185e+014 ratio=0.0263083
product(47)=6.1489e+017p2=1.58773e+016 ratio=0.0258213
product(53)=3.25892e+019 p2=7.95369e+017 ratio=0.0244059
product(59)=1.92276e+021 p2=4.30466e+019 ratio=0.0223879
product(61)=1.17288e+023 p2=2.50557e+021 ratio=0.0213625
product(67)=7.85832e+024 p2=1.5621e+023ratio=0.0198783
product(71)=5.57941e+026 p2=1.03935e+025 ratio=0.0186284
product(73)=4.07297e+028 p2=7.35601e+026 ratio=0.0180606
product(79)=3.21764e+030 p2=5.52147e+028 ratio=0.01716
product(83)=2.67065e+032 p2=4.38352e+030 ratio=0.0164137
product(89)=2.37687e+034 p2=3.67176e+032 ratio=0.0154479
product(97)=2.30557e+036 p2=3.23759e+034 ratio=0.0140425
product(101)=2.32862e+038 p2=2.99888e+036 ratio=0.0128784
product(103)=2.39848e+040 p2=2.91239e+038 ratio=0.0121427
product(107)=2.56638e+042 p2=2.96018e+040 ratio=0.0115345
product(109)=2.79735e+044 p2=3.14371e+042 ratio=0.0112382
product(113)=3.16101e+046 p2=3.483e+044 ratio=0.0110187
product(127)=4.01448e+048 p2=4.02001e+046 ratio=0.0100138
product(131)=5.25896e+050 p2=4.82701e+048 ratio=0.00917862
product(137)=7.20478e+052 p2=6.02222e+050 ratio=0.00835864
product(139)=1.00146e+055 p2=7.79739e+052 ratio=0.00778599
product(149)=1.49218e+057 p2=1.04658e+055 ratio=0.00701373
product(151)=2.2532e+059 p2=1.45467e+057 ratio=0.00645605
product(157)=3.53752e+061 p2=2.09171e+059 ratio=0.00591295
product(163)=5.76615e+063 p2=3.10866e+061 ratio=0.00539122
product(167)=9.62947e+065 p2=4.77082e+063 ratio=0.00495439
product(173)=1.6659e+068 p2=7.5543e+065ratio=0.00453467
product(179)=2.98196e+070 p2=1.23319e+068 ratio=0.00413551
product(181)=5.39735e+072 p2=2.07382e+070 ratio=0.0038423
product(191)=1.03089e+075 p2=3.59007e+072 ratio=0.00348248
product(193)=1.98962e+077 p2=6.39328e+074 ratio=0.00321331
product(197)=3.91956e+079 p2=1.17044e+077 ratio=0.00298615
product(199)=7.79992e+081 p2=2.20143e+079 ratio=0.00282238
product(211)=1.64578e+084 p2=4.25139e+081 ratio=0.0025832
product(223)=3.6701e+086 p2=8.42513e+083 ratio=0.00229561
product(227)=8.33112e+088 p2=1.71239e+086 ratio=0.00205541
product(229)=1.90783e+091 p2=3.56762e+088 ratio=0.00186999
product(233)=4.44524e+093 p2=7.61528e+090 ratio=0.00171313
product(239)=1.06241e+096 p2=1.66461e+093 ratio=0.00156682
product(241)=2.56041e+098 p2=3.72438e+095 ratio=0.0014546
product(251)=6.42663e+100 p2=8.52541e+097 ratio=0.00132658
product(257)=1.65164e+103 p2=1.99576e+100 ratio=0.00120835
product(263)=4.34383e+105 p2=4.77588e+102 ratio=0.00109946
product(269)=1.16849e+108 p2=1.16781e+105 ratio=0.000999421
product(271)=3.16661e+110 p2=2.91675e+107 ratio=0.000921098
product(277)=8.7715e+112 p2=7.43826e+109 ratio=0.000848004
product(281)=2.46479e+115 p2=1.93611e+112 ratio=0.000785507
product(283)=6.97536e+117 p2=5.14192e+114 ratio=0.000737154
product(293)=2.04378e+120 p2=1.39286e+117 ratio=0.000681513
product(307)=6.2744e+122 p2=3.84715e+119 ratio=0.000613149
product(311)=1.95134e+125 p2=1.08312e+122 ratio=0.000555067
product(313)=6.10769e+127 p2=3.10738e+124 ratio=0.000508765
product(317)=1.93614e+130 p2=9.08152e+126 ratio=0.000469053
product(331)=6.40862e+132 p2=2.70299e+129 ratio=0.000421775
product(337)=2.1597e+135 p2=8.19092e+131 ratio=0.000379261
product(347)=7.49417e+137 p2=2.52641e+134 ratio=0.000337117
product(349)=2.61547e+140 p2=7.92949e+136 ratio=0.000303177
product(353)=9.2326e+142 p2=2.53189e+139 ratio=0.000274234
product(359)=3.3145e+145 p2=8.22237e+141 ratio=0.000248072
product(367)=1.21642e+148 p2=2.71516e+144 ratio=0.000223208
product(373)=4.53726e+150 p2=9.1146e+146ratio=0.000200884
product(379)=1.71962e+153 p2=3.10976e+149 ratio=0.00018084
product(383)=6.58615e+155 p2=1.07812e+152 ratio=0.000163694
product(389)=2.56201e+158 p2=3.79717e+154 ratio=0.00014821
product(397)=1.01712e+161 p2=1.35837e+157 ratio=0.000133551
product(401)=4.07864e+163 p2=4.93462e+159 ratio=0.000120987
product(409)=1.66817e+166 p2=1.82003e+162 ratio=0.000109104
product(419)=6.98961e+168 p2=6.81408e+164 ratio=9.74887e-005
product(421)=2.94263e+171 p2=2.58916e+167 ratio=8.79881e-005
product(431)=1.26827e+174 p2=9.98282e+169 ratio=7.8712e-005
product(433)=5.49162e+176 p2=3.90491e+172 ratio=7.11068e-005
product(439)=2.41082e+179 p2=1.54938e+175 ratio=6.42677e-005
product(443)=1.06799e+182 p2=6.23471e+177 ratio=5.83778e-005
product(449)=4.79529e+184 p2=2.54399e+180 ratio=5.30519e-005
product(457)=2.19145e+187 p2=1.05241e+183 ratio=4.80234e-005
product(461)=1.01026e+190 p2=4.41318e+185 ratio=4.36838e-005
product(463)=4.67749e+192 p2=1.87565e+188 ratio=4.00995e-005
product(467)=2.18439e+195 p2=8.07824e+190 ratio=3.69817e-005
product(479)=1.04632e+198 p2=3.52519e+193 ratio=3.36912e-005
product(487)=5.09559e+200 p2=1.55842e+196 ratio=3.05837e-005
product(491)=2.50193e+203 p2=6.97851e+198 ratio=2.78925e-005
product(499)=1.24847e+206 p2=3.16486e+201 ratio=2.535e-005
product(503)=6.27978e+208 p2=1.45345e+204 ratio=2.31449e-005
product(509)=3.19641e+211 p2=6.75839e+206 ratio=2.11437e-005
product(521)=1.66533e+214 p2=3.18146e+209 ratio=1.91041e-005
product(523)=8.70967e+216 p2=1.51598e+212 ratio=1.74057e-005
product(541)=4.71193e+219 p2=7.31118e+214 ratio=1.55163e-005
product(547)=2.57743e+222 p2=3.56827e+217 ratio=1.38443e-005
product(557)=1.43563e+225 p2=1.76219e+220 ratio=1.22747e-005
product(563)=8.08258e+227 p2=8.80475e+222 ratio=1.08935e-005
product(569)=4.59899e+230 p2=4.45045e+225 ratio=9.67702e-006
product(571)=2.62602e+233 p2=2.27542e+228 ratio=8.6649e-006
product(577)=1.51521e+236 p2=1.17664e+231 ratio=7.7655e-006
product(587)=8.89431e+238 p2=6.15319e+233 ratio=6.91813e-006
product(593)=5.27432e+241 p2=3.25377e+236 ratio=6.16907e-006
product(599)=3.15932e+244 p2=1.73963e+239 ratio=5.50634e-006
product(601)=1.89875e+247 p2=9.40295e+241 ratio=4.95218e-006
product(607)=1.15254e+250 p2=5.13767e+244 ratio=4.45769e-006
product(613)=7.06508e+252 p2=2.83739e+247 ratio=4.01608e-006
product(617)=4.35916e+255 p2=1.58373e+250 ratio=3.63311e-006
product(619)=2.69832e+258 p2=8.93323e+252 ratio=3.31067e-006
product(631)=1.70264e+261 p2=5.09169e+255 ratio=2.99047e-006
product(641)=1.09139e+264 p2=2.93225e+258 ratio=2.68671e-006
product(643)=7.01765e+266 p2=1.70603e+261 ratio=2.43105e-006
product(647)=4.54042e+269 p2=1.00272e+264 ratio=2.20842e-006
product(653)=2.96489e+272 p2=5.95303e+266 ratio=2.00784e-006
product(659)=1.95386e+275 p2=3.56968e+269 ratio=1.82699e-006
product(661)=1.2915e+278 p2=2.1618e+272ratio=1.67386e-006
product(673)=8.69182e+280 p2=1.32208e+275 ratio=1.52106e-006
product(677)=5.88436e+283 p2=8.16439e+277 ratio=1.38747e-006
product(683)=4.01902e+286 p2=5.09068e+280 ratio=1.26665e-006
product(691)=2.77714e+289 p2=3.20465e+283 ratio=1.15394e-006 这个应该没有通式解吧。素数定理π(N)~N/ln(N)，到现在还没有证明呢。要证明了“黎曼猜想”，那就是数学界的大事了。 在维基百科上找到了这个题目，条目名称是《质数阶乘》
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