Let $Y(k)$ be the number consisting of $1$, repeated $k$ times. We know that $Y(2) =11$ is prime. It so happens that $Y(19)$ and $Y(23)$ are also prime.
Are there any more?
Regards,
David
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J. W. Tanner
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David Sycamore
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1Do you mean prime numbers with digits all $1$ ? Google rep-units. The next is $Y(317)$ – Peter Mar 06 '19 at 22:36
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4Y(n) is prime for n = 2, 19, 23, 317, 1031, ... (sequence A004023 in OEIS). – J. W. Tanner Mar 06 '19 at 22:36
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2The magic words are "Repunit primes". There are five known repunit primes. http://oeis.org/A004023 – Arturo Magidin Mar 06 '19 at 22:37
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@ArturoMagidin and at least one prp – Peter Mar 06 '19 at 22:37
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1These numbers are called
"repunits." If you search this site for "prime repunit" you'll get lots of hits. – B. Goddard Mar 06 '19 at 22:38 -
1May (or may not) be worth noting a repunit $Y(k)$ is prime only if $k$ is. But the converse is obviously not true. – fleablood Mar 07 '19 at 00:18
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@ J. W. Tanner This is not to suggest that we know of any more after 1031 or that there are infinitely many of them I presume. – DDS Jun 26 '19 at 18:34
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Richard K. Guy in the 2nd edition of his "Unsolved Problems in Number Theory'' also cites Y(49081) as "probably'' one. – DDS Jun 26 '19 at 18:40
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per OEIS, Y(n) is probably prime for $n=49081, 86453, 109297, 270343, 5794777, $ and $8177207$ – J. W. Tanner Sep 30 '21 at 18:32
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To summarize the comments, the repunit $$Y(k)=\frac{10^k-1}9,$$ the number $11...111$ consisting of $k$ copies of the digit $1,$ is known to be prime for
$k=2, 19, 23, 317,$ and $1031.$
J. W. Tanner
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Thanks to everyone who replied; much appreciated, now I know what a repunit prime is! I came across these when looking at something else, namely: – David Sycamore Mar 07 '19 at 13:01
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Let P(n,k) be the number Y(k)prime(n)Y(k) and sequence a(n) be least k>0 such that P(n,k) is prime, and 0 if no such k exists. I find the following: 0,1,1,3,0,3,1,0,1,1,2,0,3,2,1,0,14,3,0,3,2,1,4,3,0.... The problem is with the zeroes because it is hard to prove that no such k exists. Therefore, though I am certain of the non zero terms the zeroes are best guesses at the moment. I would like to submit this to oeis when and if more is known. Any ideas for proofs most welcome, though the case for prime(5)=11 looks obvious. – David Sycamore Mar 07 '19 at 13:27
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It so happens that Y(19) is the greatest prime divisor of P(2,18), and Y(23) is the greatest prime divisor of P(2,22), which is how I came across them in the first place. – David Sycamore Mar 07 '19 at 13:38
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1???What vote? I have not (to my knowledge) voted on anything. Was very happy with responses to my first question (thanks). Have tried to ask a follow up question but no replies as yet. – David Sycamore Mar 09 '19 at 10:03
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Since this was posted, $Y(k)$ has also shown to be prime for $k=49081$, $86453$ – J. W. Tanner Sep 29 '24 at 01:56