Prime Question n||p

[Carl G.]

I had the following thought when I was trying to fall asleep the other day:

The prime numbers 2 and 5 have the property that when a number is formed
by concatenating an integer from 1 to infinity with the prime, the
number is always composite (for 2: 12, 22, 32, 42, ... 102, 112, ...;
and for 5: 15, 25, 35, 45, ...). Using "||" as a concatenation
operator, then this can be expressed as: If p is the prime and n is in
the set of integers from 1 to infinity, then n||p is composite. For
most primes, some of the numbers in the set formed by concatenation
would be composite and some would be prime. For example, for 11: 111 is
composite, 211 is prime, 311 is prime, 411 is composite, etc. Are there
primes other than 2 and 5 in which all the numbers would be composite?

[Gareth Taylor]

Are there primes other than 2 and 5 in which all the numbers would be
composite?

No, by Dirichlet's theorem on primes in arithmetic progression, which
says that if a and d are coprime integers then there are infinitely many
primes of the form a+nd.

Your sequences have this form: e.g., 111, 211, 311, 411, ... is 11+100n.

Any prime other than 2 or 5 is coprime to that 10^k term, and so the
sequence will have many primes in it.

https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions

Gareth

[Doc O'Leary ,]

For your reference, records indicate that

Post by Carl G.
The prime numbers 2 and 5 have the property that

they are factors of 10. I think anything beyond that is numerology.

It’s like realizing that the digits for multiples of 9 add up to multiples of 9.

[Richard Tobin]

Post by Doc O'Leary ,

Post by Carl G.
The prime numbers 2 and 5 have the property that

they are factors of 10. I think anything beyond that is numerology.

That explains why they have that property, but shows nothing about
the case for other digits, which is a much more interesting problem.

-- Richard

[Doc O'Leary ,]

For your reference, records indicate that

Post by Richard Tobin

Post by Doc O'Leary ,

Post by Carl G.
The prime numbers 2 and 5 have the property that

they are factors of 10. I think anything beyond that is numerology.

That explains why they have that property, but shows nothing about
the case for other digits, which is a much more interesting problem.

I disagree. There’s nothing to “show” for other digits. It’s just math,
and/or a quirk of our common base-10 representation. I mean, feel free
to explore a 3 * 7 = base-21 system to see what “interesting” things may
hold true. Nothing wrong with finding new ways to count sheep. :-)

[Gareth Taylor]

I disagree. There’s nothing to “show” for other digits. It’s just
math, and/or a quirk of our common base-10 representation. I mean,
feel free to explore a 3 * 7 = base-21 system to see what
“interesting” things may hold true. Nothing wrong with finding new
ways to count sheep. :-)

It may be just maths, but it's interesting and challenging maths! For
all primes other than 2 or 5, the sequence described contains infinitely
many primes and infinitely many composites.

Yes, the "other than 2 or 5" bit is to do with our number base being 10.
If you worked in base 21 then it would be "other than 3 or 7".

As for something to show, it's answering what the original question
asked.

Gareth