58 lines
2.7 KiB
Text
58 lines
2.7 KiB
Text
A recent discovery, quoted from [https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/ Quantamagazine] (March 13, 2016):
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'' Two mathematicians have uncovered a simple, previously unnoticed property of ''
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'' prime numbers — those numbers that are divisible only by 1 and themselves. ''
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'' Prime numbers, it seems, have decided preferences about the final digits of ''
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'' the primes that immediately follow them.
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and
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'' This conspiracy among prime numbers seems, at first glance, to violate a ''
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'' longstanding assumption in number theory: that prime numbers behave much ''
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'' like random numbers.
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'' ─── (original authors from Stanford University): ''
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'' ─── Kannan Soundararajan and Robert Lemke Oliver ''
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The task is to check this assertion, modulo 10.
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Lets call <big><code> i -> j </code></big> a transition if <big><code> i </code></big> is the last decimal digit of a prime, and <big><code> j </code></big> the last decimal digit of the following prime.
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;Task:
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Considering the first one million primes. Count, for any pair of successive primes, the number of transitions <big><code> i -> j </code></big> and print them along with their relative frequency, sorted by <big><code> i </code>.</big>
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You can see that, for a given <big><code> i </code>,</big> frequencies are not evenly distributed.
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;Observation:
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(Modulo 10), primes whose last digit is '''9''' "prefer" the digit '''1''' to the digit '''9''', as its following prime.
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;Extra credit:
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Do the same for one hundred million primes.
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;Example for 10,000 primes:
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<pre>
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10000 first primes. Transitions prime % 10 → next-prime % 10.
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1 → 1 count: 365 frequency: 3.65 %
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1 → 3 count: 833 frequency: 8.33 %
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1 → 7 count: 889 frequency: 8.89 %
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1 → 9 count: 397 frequency: 3.97 %
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2 → 3 count: 1 frequency: 0.01 %
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3 → 1 count: 529 frequency: 5.29 %
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3 → 3 count: 324 frequency: 3.24 %
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3 → 5 count: 1 frequency: 0.01 %
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3 → 7 count: 754 frequency: 7.54 %
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3 → 9 count: 907 frequency: 9.07 %
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5 → 7 count: 1 frequency: 0.01 %
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7 → 1 count: 655 frequency: 6.55 %
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7 → 3 count: 722 frequency: 7.22 %
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7 → 7 count: 323 frequency: 3.23 %
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7 → 9 count: 808 frequency: 8.08 %
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9 → 1 count: 935 frequency: 9.35 %
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9 → 3 count: 635 frequency: 6.35 %
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9 → 7 count: 541 frequency: 5.41 %
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9 → 9 count: 379 frequency: 3.79 %
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</pre>
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<br><br>
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