Primes of the form .$..55555444433322122333444455555...$

Asked Aug 08 '15 at 16:40

Active Aug 08 '15 at 16:44

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What is the smallest prime number of the form $...55555444433322122333444455555...$, where the concatenation runs through the first natural numbers, and where each decimal number $n$ being repeated/written $n$ times, for example: $$13131313131313131313131313121212121212121212121212....444433322122333444\\ ....12121212121212121212121213131313131313131313131313$$

(Lets call it $n = 13$)

I've checked to $n = 19$. Is there prime number of such form ?

edited Aug 08 '15 at 16:44

Michael Galuza

4,730

asked Aug 08 '15 at 16:40

Kazuya Kazunaga

If there is, $n$ must be of the form $3k+1$ (or your number will be a multiple of $3$). – Micah Aug 08 '15 at 16:45
What base are you using? – barak manos Aug 08 '15 at 16:46
@barakmanos, I used base 10 – Kazuya Kazunaga Aug 08 '15 at 16:46
What's so special about $10$ in the context of this question? – barak manos Aug 08 '15 at 16:47
@barakmanos perhaps it is the system we are all most familiar with? – Conor O'Brien Aug 08 '15 at 16:48
@CᴏɴᴏʀO'Bʀɪᴇɴ: In the context of this question. – barak manos Aug 08 '15 at 16:49
1

Well in that case $22122_3=233_{10}$ is prime. – barak manos Aug 08 '15 at 16:50
@barakmanos alright, alright, no need to bold at me ._. – Conor O'Brien Aug 08 '15 at 16:52
@CᴏɴᴏʀO'Bʀɪᴇɴ: hahaha. – barak manos Aug 08 '15 at 16:53
@barakmanos, Thanks, but here we are interested in base10 : ) – Kazuya Kazunaga Aug 08 '15 at 16:53
1

@KazuyaKazunaga: OK, I see you've changed your mind. I personally think that there is nothing interesting in particular with base $10$ in the context of this question, but perhaps somebody will come up with an answer for this specific case. – barak manos Aug 08 '15 at 16:56
n >= 721, with the resulting value at least 1.5 million digits. – DanaJ Aug 09 '15 at 16:46
n >= 873, with over 2.2 million digits. No more searching for me. – DanaJ Aug 26 '15 at 05:55

Primes of the form .$..55555444433322122333444455555...$

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