Prime conspiracy
Prime Conspiracy
A recent discovery, quoted from Quantamagazine , Marsh 23,2016 : "A previously unnoticed property of prime numbers seems to violate a longstanding assumption about how they behave. Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them."
The task is to check this assertion, modulo 10. Lets call i -> j a transition if i is the last decimal digit of a prime, and j the last decimal digit of the following prime.
Task
Considering the first 1_000_000 primes, count the number of transitions i -> j , print them along with their relative frequency. i and j are in (1,3,7,9) . Check that, for a given i, frequencies are not evenly distributed.
Extra credit : The same for 100_000_000 primes.
Example>>.words
10000 first primes. Transitions prime % 10 → next-prime % 10. 1 → 1 count: 365 frequency: 3.65 % 1 → 3 count: 833 frequency: 8.33 % 1 → 7 count: 889 frequency: 8.89 % 1 → 9 count: 397 frequency: 3.97 % 3 → 1 count: 529 frequency: 5.29 % 3 → 3 count: 324 frequency: 3.24 % 3 → 7 count: 754 frequency: 7.54 % 3 → 9 count: 907 frequency: 9.07 % 7 → 1 count: 655 frequency: 6.55 % 7 → 3 count: 722 frequency: 7.22 % 7 → 7 count: 323 frequency: 3.23 % 7 → 9 count: 808 frequency: 8.08 % 9 → 1 count: 935 frequency: 9.35 % 9 → 3 count: 636 frequency: 6.36 % 9 → 7 count: 541 frequency: 5.41 % 9 → 9 count: 379 frequency: 3.79 %
Perl 6
<lang perl6>my @primes := 3, |(flat map( { $^i-1, $i+1 }, (6, 12, 18 ... *))).grep: *.is-prime;
my %conspiracy;
my $previous = 2;
my $upto = 1_000_000;
map { my $l = $_ % 10; %conspiracy{"$previous → $l count:"}++; $$previous = $l }, @primes[^$upto];
say "$_ \tfrequency: {($_.value/$upto *100).round(.01)} %" for %conspiracy.sort;</lang>
- Output:
1 → 1 count: 42853 frequency: 4.29 % 1 → 3 count: 77475 frequency: 7.75 % 1 → 7 count: 79453 frequency: 7.95 % 1 → 9 count: 50153 frequency: 5.02 % 2 → 3 count: 1 frequency: 0 % 3 → 1 count: 58255 frequency: 5.83 % 3 → 3 count: 39668 frequency: 3.97 % 3 → 5 count: 1 frequency: 0 % 3 → 7 count: 72828 frequency: 7.28 % 3 → 9 count: 79358 frequency: 7.94 % 5 → 7 count: 1 frequency: 0 % 7 → 1 count: 64230 frequency: 6.42 % 7 → 3 count: 68595 frequency: 6.86 % 7 → 7 count: 39603 frequency: 3.96 % 7 → 9 count: 77586 frequency: 7.76 % 9 → 1 count: 84596 frequency: 8.46 % 9 → 3 count: 64371 frequency: 6.44 % 9 → 7 count: 58130 frequency: 5.81 % 9 → 9 count: 42843 frequency: 4.28 %