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Mike Keith found that the decimal fraction 40999920000041/(999999^3) has the value 0.000 41 000 43 000 47 000 53 000 61 000 71 000 83 000 97 00 113 00 131 00 151 00 173 00 197 00 223 00 251 00 281 00 313 00 347 00 383 00 421 00 461 00 503 00 547 00 593 00 641 00 691 00 743 00 797 00 853 00 911 00 971 0 1033 0 1097 0 1163 0 1231 0 1301 0 1373 0 1447 0 1523 0 1601 ... Note the 40 prime numbers amongst the zeros! The first 40 digits of Euler's constant form a prime. [Kulsha] The expression P(n) = n^2 + n + 41 is prime for all natural numbers n < 40. It is composite for n = 40. [Mitchell] The sequence number for prime numbers is A000040 in The OnLine Encyclopedia of Integer Sequences. [Hartley] 40 = 2^3 * 5 (the first three primes in order). [Beech] 40!+39!+38!+...+3!+2!+1!+0!+1!+2!+3!+...+38!+39!+40! is prime. [Schiffman]
There are 40 permissible 2digit endings for prime numbers. Consider the following "race" after the first 1000 primes:
Clearly, we have "Car Number 57" in the lead with 29 primes after 1000 laps (primes). But what about after 10000 laps? 100000 laps? 10^25 laps? Place your bets now!Let a(n) = the number of primes that end with "01" among the first 10^n primes. The sequence begins 0, 2, 25, ... Let a(n) = the number of primes that end with "03" among the first 10^n primes. The sequence begins 0, 2, 25, ... Let a(n) = the number of primes that end with "07" among the first 10^n primes. The sequence begins 0, 2, 27, ... Let a(n) = the number of primes that end with "09" among the first 10^n primes. The sequence begins 0, 3, 24, ... Let a(n) = the number of primes that end with "11" among the first 10^n primes. The sequence begins 1, 3, 25, ... Let a(n) = the number of primes that end with "13" among the first 10^n primes. The sequence begins 1, 3, 23, ... Let a(n) = the number of primes that end with "17" among the first 10^n primes. The sequence begins 1, 2, 25, ... Let a(n) = the number of primes that end with "19" among the first 10^n primes. The sequence begins 1, 2, 27, ... Let a(n) = the number of primes that end with "21" among the first 10^n primes. The sequence begins 0, 2, 27, ... Let a(n) = the number of primes that end with "23" among the first 10^n primes. The sequence begins 1, 3, 28, ... Let a(n) = the number of primes that end with "27" among the first 10^n primes. The sequence begins 0, 2, 23, ... Let a(n) = the number of primes that end with "29" among the first 10^n primes. The sequence begins 1, 2, 26, ... Let a(n) = the number of primes that end with "31" among the first 10^n primes. The sequence begins 0, 4, 24, ... Let a(n) = the number of primes that end with "33" among the first 10^n primes. The sequence begins 0, 2, 24, ... Let a(n) = the number of primes that end with "37" among the first 10^n primes. The sequence begins 0, 3, 25, ... Let a(n) = the number of primes that end with "39" among the first 10^n primes. The sequence begins 0, 3, 23, ... Let a(n) = the number of primes that end with "41" among the first 10^n primes. The sequence begins 0, 3, 21, ... Let a(n) = the number of primes that end with "43" among the first 10^n primes. The sequence begins 0, 2, 24, ... Let a(n) = the number of primes that end with "47" among the first 10^n primes. The sequence begins 0, 2, 26, ... Let a(n) = the number of primes that end with "49" among the first 10^n primes. The sequence begins 0, 3, 23, ... Let a(n) = the number of primes that end with "51" among the first 10^n primes. The sequence begins 0, 2, 28, ... Let a(n) = the number of primes that end with "53" among the first 10^n primes. The sequence begins 0, 2, 27, ... Let a(n) = the number of primes that end with "57" among the first 10^n primes. The sequence begins 0, 3, 29, ... Let a(n) = the number of primes that end with "59" among the first 10^n primes. The sequence begins 0, 2, 26, ... Let a(n) = the number of primes that end with "61" among the first 10^n primes. The sequence begins 0, 2, 23, ... Let a(n) = the number of primes that end with "63" among the first 10^n primes. The sequence begins 0, 3, 23, ... Let a(n) = the number of primes that end with "67" among the first 10^n primes. The sequence begins 0, 4, 25, ... Let a(n) = the number of primes that end with "69" among the first 10^n primes. The sequence begins 0, 1, 22, ... Let a(n) = the number of primes that end with "71" among the first 10^n primes. The sequence begins 0, 2, 23, ... Let a(n) = the number of primes that end with "73" among the first 10^n primes. The sequence begins 0, 3, 27, ... Let a(n) = the number of primes that end with "77" among the first 10^n primes. The sequence begins 0, 1, 25, ... Let a(n) = the number of primes that end with "79" among the first 10^n primes. The sequence begins 0, 4, 26, ... Let a(n) = the number of primes that end with "81" among the first 10^n primes. The sequence begins 0, 2, 24, ... Let a(n) = the number of primes that end with "83" among the first 10^n primes. The sequence begins 0, 3, 27, ... Let a(n) = the number of primes that end with "87" among the first 10^n primes. The sequence begins 0, 1, 23, ... Let a(n) = the number of primes that end with "89" among the first 10^n primes. The sequence begins 0, 2, 25, ... Let a(n) = the number of primes that end with "91" among the first 10^n primes. The sequence begins 0, 2, 25, ... Let a(n) = the number of primes that end with "93" among the first 10^n primes. The sequence begins 0, 2, 24, ... Let a(n) = the number of primes that end with "97" among the first 10^n primes. The sequence begins 0, 3, 25, ... Let a(n) = the number of primes that end with "99" among the first 10^n primes. The sequence begins 0, 2, 24, ... Proposed by G. L. Honaker, Jr. (June 2016)
(There are 2 curios for this number that have not yet been approved by an editor.)
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