Problems & Puzzles: Puzzles

 Puzzle 416. Prime numbers less than 2^18 Patrick Capelle points out the following Curio by Salamin: There are exactly 23000 prime numbers less than 218. Patrick asks: Q1. Are there other powers nm such that  π(nm) = a * 10^b, where a and b>0 are natural numbers?

Contributions came from Dan Dima, Jacques Tramu, Enoch Haga & Farideh Firoozbakht

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Farideh wrote:

There exist many such powers.

The best solutions that I found are :

For m=2  and b=5:
1. pi( 27608^2)=393*10^5
2. pi(44716,^2)=982*10^5

For m=3  and b=3:
1. pi(64^3)=pi(2^18)=23*10^3
2. pi(941^3)=42788*10^3
3. pi(1253^3)=96691*10^3.

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Enoch wrote:

1900 primes under 2^14 -- 1900=19*10^2

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Tramu wrote:

Some n,m, n**m, pi(n**m)  such as pi(n**m)  mod  10 == 0

2       14      16384   1900
2       18      262144  23000
2       19      524288  43390
2       29      536870912       28192750
3       15      14348907        931260
3       22      31381059609     1357200840
5       3       125     30
5       15      30517578125     1321453490
6       4       1296    210
11      2       121     30
12      5       248832  21950
12      6       2985984 215880
14      3       2744    400
14      8       1475789056      73579310
23      6       148035889       8340120
26      3       17576   2020
28      3       21952   2460
29      1       29      10
30      1       30      10
35      2       1225    200
38      6       3010936384      144950410
39      2       1521    240
41      3       68921   6850
43      4       3418801 244810
45      3       91125   8810
45      5       184528125       10268090
52      3       140608  13060
57      6       34296447249     1477597400
61      5       844596301       43319080
62      5       916132832       46792180
63      3       250047  22050
66      3       287496  25030
70      3       343000  29420
71      1       71      20
72      1       72      20
72      2       5184    690
72      3       373248  31770
74      5       2219006624      108423020
87      2       7569    960
87      4       57289761        3410690
89      2       7921    1000
91      3       753571  60500
92      3       778688  62370
94      2       8836    1100

2       39      549755813888    21151907950
7       14      678223072849    25885182840
13      10      137858491849    5602858280
18      9       198359290368    7944039180
31      8       852891037441    32268888030
93      6       646990183449    24737729430
94      6       689869781056    26312553920

***

Dan Dima wrote:

primepi(2^14) = 1900
primepi(2^18) = 23000
primepi(5^3) = 30
primepi(6^4) = 210
primepi(11^2) = 30
primepi(12^5) = 21950
primepi(14^3) = 400
primepi(26^3) = 2020
primepi(28^3) = 2460
primepi(35^2)=200
primepi(39^2)=240
primepi(41^3)=6850
primepi(45^3)=8810
primepi(52^3)=13060
primepi(63^3)=22050
primepi(66^3)=25030
primepi(70^3)=29420
primepi(72^2)=690
primepi(72^3)=31770
primepi(87^2)=960
primepi(89^2)=1000
primepi(94^2)=1100
primepi(108^2)=1400
primepi(139^2)=2190
primepi(143^2)=2310
primepi(152^2)=2580
primepi(165^2)=2980
primepi(166^2)=3010
primepi(207^2)=4480
primepi(234^2)=5570
primepi(251^2)=6320
primepi(264^2)=6910
primepi(265^2)=6960
primepi(272^2)=7300
primepi(275^2)=7450
primepi(279^2)=7650
primepi(285^2)=7950
primepi(298^2)=8600
primepi(303^2)=8870
primepi(310^2)=9260
primepi(316^2)=9580
primepi(334^2)=10580
primepi(336^2)=10690
primepi(349^2)=11460
primepi(366^2)=12490
primepi(389^2)=13960
primepi(391^2)=14100
primepi(395^2)=14360
primepi(396^2)=14420
primepi(397^2)=14490
primepi(420^2)=16030
primepi(468^2)=19530
primepi(489^2)=21150
primepi(491^2)=21310
primepi(493^2)=21470
primepi(521^2)=23760
primepi(524^2)=24000
primepi(533^2)=24760
primepi(548^2)=26020
primepi(551^2)=26290
primepi(552^2)=26380
primepi(557^2)=26820
primepi(568^2)=27820
primepi(592^2)=30010
primepi(602^2)=30940
primepi(622^2)=32830
primepi(633^2)=33910
primepi(653^2)=35880
primepi(668^2)=37430
primepi(674^2)=38030
primepi(683^2)=38950

Certainly there are an infinite number of such powers and we expect in
fact that a fraction of 1/10 of all n^m powers should have this
property.

***

J. K. Andersen wrote (Jan., 2009):

The first 6 squares giving b=6 ending zeroes:
pi(1179629^2) = 51689000000
pi(1260897^2) = 58765000000
pi(1842967^2) = 122112000000
pi(2187095^2) = 169878000000
pi(2496065^2) = 219205000000
pi(2748297^2) = 263955000000

Below is the largest found number of ending zeroes for each exponent.
If more than one case was found for that exponent and number of zeroes
then only the smallest found case is listed.
pi(1179629^2) = 51689000000
pi(22107^3) = 372893900000
pi(1106^4) = 55431460000
pi(1230^5) = 81501427346000
pi(8^6) = pi(2^18) = 23000
pi(4^7) = pi(2^14) = 1900
pi(46^8) = 677450454800
pi(4^9) = pi(2^18) = 23000
pi(13^10) = 5602858280
pi(9^11) = pi(3^22) = 1357200840
pi(14^12) = 1850708502230
pi(8^13) = pi(2^39) = 21151907950
pi(2^14) = 1900
pi(3^15) = 931260
pi(7^16) = 1104127111380
pi(9^17) = pi(3^34) = 459127968423610
pi(2^18) = 23000
pi(2^19) = 43390
pi(8^20) = pi(2^60) = 28423094496953330

pi(3^22) = 1357200840
pi(8^23) = pi(2^69) = 12611864618760352880, the smallest for 23
if pi(7^23) does not end in 0.
pi(4^24) = pi(2^48) = 8731188863470
pi(4^28) = pi(2^56) = 1906879381028850
pi(2^29) = 28192750
pi(4^30) = pi(2^60) = 28423094496953330
pi(3^34) = 459127968423610
pi(2^39) = 21151907950
pi(2^48) = 8731188863470
pi(2^56) = 1906879381028850
pi(2^58) = 7357400267843990
pi(2^59) = 14458792895301660
pi(2^60) = 28423094496953330
pi(2^69) = 12611864618760352880

Here are a few large powers of 10 giving one 0:
pi(10^18) = 24739954287740860
pi(10^20) = 2220819602560918840
pi(10^22) = 201467286689315906290

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