Problems & Puzzles:
Perhaps you have seen this
curio about the prime
which cannot become another by inserting a digit (neither
appending it at the extremes, CR). [Blanchette]
It's a kind of easier to
get a few more of them:
369293, 5364431, 7904521, 9387527, 9510341, 22038829, 39113161,
43744697, 56618161, 57640663, ...
But I intuit that there
must be only a finite quantity cases of this type.
1. Can you argument on
favor or on the contrary of my intuition?
2. If my intuition is
correct can you find the largest case of this type of primes?
Contributions came from Giovanni Resta, J.K.Andersen, Adam
Stinchcombe & Farideh Firoozbakht.
All of them demonstrated that my intuition was wrong, using more or
less the same argument:
Consider a prime p with n-1 digits. There are approximately 10n
insert a digit in p and get an n-digit number ending in 1, 3, 7 or
A random n-digit number has chance around 1/log(10^n) of being
prime. When it is not divisible by 2 or 5, this becomes 2 * 5/4 *
2.5/(log(10)*n) ~= 1.1/n. If we assume the 10n required composites
randomly for numbers coprime to 10, then the heuristic chance of all
composite is around (1-1.1/n)^(10n). This approaches a number near
1/60000 for large n, so I expect a positive ratio of primes to have
Andersen adds that:
unpublished currently) there are infinitely many composite numbers
to 10 with this property. I can imagine how it could be proved for
composites by finding a covering set of prime factors. A proof for
primes seems much harder. I can imagine covering sets which show
there are infinitely many numbers of forms that are likely to
include infinitely many primes. But I guess it would be very hard to
prove whether such forms really have infinitely many primes. I have
not searched for such forms.
All noticed that my list of small cases was not complete:
primes with this property. The list in the puzzle is non-exhaustive.
The largest example gotten by some of them was:
10^999 + 33592689 (Resta)