The previous post showed how to list all limerick primes. This post shows how to list all sonnet primes. These are primes of the form ababcdcdefefgg, the rhyme scheme of an English (Shakespearean) sonnet, where the letters a through g represent digits and a is not zero.
Here’s the Mathematica code.
number[s_] := 10100000000000 s[[1]] + 1010000000000 s[[2]] +
1010000000 s[[3]] + 101000000 s[[4]] +
101000 s[[5]] + 10100 s[[6]] + 11 s[[7]]
test[n_] := n > 10^13 && PrimeQ[n]
candidates = Permutations[Table[i, {i, 0, 9}], {7}];
list = Select[Map[number, candidates], test];
The function Permutations[list, {n}] creates a list of all permutations of list of length n. In this case we create all permutations the digits 0 through 9 that have length 7. These are the digits a through g.
The function number turns the permutation into a number. This function is applied to each candidate. We then select all 14-digit prime numbers from the list of candidates using test.
If we ask for Length[list] we see there are 16,942 sonnet primes.
As mentioned before, the smallest of these primes is 10102323454577.
The largest is 98987676505033.
Related post: Limerick primes
Seemed natural to work in base 7 instead, but of course odd bases are no good. In base 8, there are 1057 solutions, all of which use the digit 0. So, 10 is the first base where there’s a sonnet prime that doesn’t include a 0 digit (and in fact there are 6367 of those.
Still obsessed with base 7. All the sonnet numbers are multiples of 6 there, but 204 of them are of the form 6 times a prime.
Hey John — here’s a real he-man prime challenge:
Find a non-trivial pattern (p1, p2, … pn) such that the size of the set of primes with that pattern is itself a member of the set!
I bet this is impossible without allowing leading zeros to be part of the pattern — since I seem to recall that the total number of primes with N digits is not between 10^(N-1) and (10^N)-1.