Primes of the form 2...21

Asked Jun 16 '24 at 04:59

Active Jun 16 '24 at 04:59

Viewed 89 times

I was wondering what properties could have these numbers: $21, 221, 2221, 22221, ...$ At glance I thought this set would have infinitely many primes. Immediately I went to Python and I realized that from $21$ to $2...21$ (300 digits) the only prime is $2221$.

Therefore my next question was: is there at least one more prime? Or are them all composite?

This was my approach: $$N=1+20\sum_{i=0}^k 10^i$$

If $$k \equiv 0 \bmod{3} \rightarrow N=21, 22221, ... $$ $N$ is divisible by $3$ and it's not a prime.

If not, I have no clue. But most of them end up being divisible by a small prime like $7$, $13$ or $19$.

It seems such a trivial quiestion but so hard to answer. Does anyone have an approach to the problem? Thank you.

asked Jun 16 '24 at 04:59

Francisco J. Maciel Henning

1

Isn't the 16th one: $222,222,222,222,222,221$ prime? – Eric Towers Jun 16 '24 at 06:02
1

It looks like the 98th, 119th, and 242nd are also prime... What method of detecting primes did you use? – Eric Towers Jun 16 '24 at 06:04
See also $,33\ldots 331$ in this question. $\ \ $ – Bill Dubuque Jun 16 '24 at 06:26
1

The first primes of this form : factordb , another two primes factordb – Peter Jun 16 '24 at 07:43
1

((1)) Your Python Script is wrong : It might be handling very large integers wrongly , having round-off errors during division , which might make Primes look like Composites. ((2)) In general , Arithmetic Progressions having at least 1 Prime might have infinite number of Primes. – Prem Jun 16 '24 at 07:44
@Prem This thumb rule is however only "reliable" , if we find one "large" prime. – Peter Jun 16 '24 at 07:46
1

As said , probably infinite many of those numbers are prime , but a proof is almost surely out of reach. – Peter Jun 16 '24 at 07:50
2

Instead of Python , you should use PARI/GP for this purpose. – Peter Jun 16 '24 at 07:50
1

Finally , why do you consider this to be trivial ? We have nothing better than to test the numbers (after sieving out cases with small factors). Such questions are not trivial at all. – Peter Jun 16 '24 at 07:53
Thank you for the answers! I did not know Python would fail with large integers. Appreciate them. – Francisco J. Maciel Henning Jun 16 '24 at 15:52
@FranciscoJavierMacielHennin See if python has a library which invokes thing like "gmp" , a library in itself which handles integers of arbitrary size. https://gmplib.org/ – Carl Witthoft Jun 16 '24 at 18:09
See https://oeis.org/A084832 – Hugo Pfoertner Jun 19 '24 at 16:53
Thank you for the feedback! I will check this libraries – Francisco J. Maciel Henning Jun 22 '24 at 23:57
Recursive is a(n+1)=10*a(n)+11 – Roddy MacPhee Apr 02 '25 at 00:24
https://en.m.wikipedia.org/wiki/Iterated_function might help use x=a(n) and we get a^n x +... mod p – Roddy MacPhee Apr 02 '25 at 00:31

Primes of the form 2...21

0 Answers0