# Primes from the sum of the first n repunits $1+11+111+1111+11111+...$

Is there prime number of the form $$1+11+111+1111+11111+...$$. I've checked it up to first 2000 repunits, but i found none. If $$R_1=1$$, $$R_2=1+11$$, $$R_3=1+11+111$$, $$R_n=1+11+111+...+$$nth repunit. What is the smallest n such that $$R_n$$ is prime ?

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Here's the smallest such prime, found using this code:

Select[Accumulate[Table[
Sum[10^i, {i, 0, n}],
{n, 0, 10000}]], PrimeQ]


$$1234567901234567901234567901234567901234567901234567901234567901234567 9012345679012345679012345679012345679012345679012345679012345679012345 6790123456790123456790123456790123456790123456790123456790123456790123 4567901234567901234567901234567901234567901234567901234567901234567901 2345679012345679012345679012345679012345679012345679012345679012345679 0123456790123456790123456790123456790123456790123456790123456790123456 7901234567901234567901234567901234567901234567901234567901234567901234 5679012345679012345679012345679012345679012345679012345679012345679012 3456790123456790123456790123456790123456790123456790123456790123456790 1234567901234567901234567901234567901234567901234567901234567901234567 9012345679012345679012345679012345679012345679012345679012345679012345 6790123456790123456790123456790123456790123456790123456790123456790123 4567901234567901234567901234567901234567901234567901234567901234567901 2345679012345679012345679012345679012345679012345679012345679012345679 0123456790123456790123456790123456790123456790123456790123456790123456 7901234567901234567901234567901234567901234567901234567901234567901234 5679012345679012345679012345679012345679012345679012345679012345679012 3456790123456790123456790123456790123456790123456790123456790123456790 1234567901234567901234567901234567901234567901234567901234567901234567 9012345679012345679012345679012345679012345679012345679012345679012345 6790123456790123456790123456790123456790123456790123456790123456790123 4567901234567901234567901234567901234567901234567901234567901234567901 2345679012345679012345679012345679012345679012345679012345679012345679 0123456790123456790123456790123456790123456790123456790123456790123456 7901234567901234567901234567901234567901234567901234567901234567901234 5679012345679012345679012345679012345679012345679012345679012345679012 3456790123456790123456790123456790123456790123456790123456790123456790 1234567901234567901234567901234567901234567901234567901234567901234567 9012345679012345679012345679012345679012345679012345679012345679012345 6790123456790123456790123456790123456790123456790123456790123456790123 4567901234567901234567901234567901234567901234567901234567901234567901 2345679012345679012345679012345679012345679012345679012345679012345679 0123456790123456790123456790123456790123456790123456790123456790123456 7901234567901234567901234567901234567901234567901234567901234567901234 5679012345679012345679012345679012345679012345679012345679012345679012 34567901234567901234567901234567901234567900957$$

The first one is at $$n=2497$$, you just missed it!

Do[If[PrimeQ[(10^(n+1)-9n-10)/81], Print[n]], {n, 10^4}]
(* 2497 *)
(* 3301 *)
...


The direct formula for the $$n^{\text{th}}$$ term is from

Sum[10^j, {i, 0, n - 1}, {j, 0, i}]
(* 1/81 (-10 + 10^(1 + n) - 9 n) *)


To give some more examples of solutions, using Roman's direct formula for conciseness (alternately you can do something like r = 10*r+1; N += r in the loop).

Pari/GP (test is AES BPSW):

? for(n=1,6000,ispseudoprime((10^(n+1)-9*n-10)/81) && print(n))
2497
3301
time = 2min, 5,916 ms.


Perl/ntheory (v0.74) (test is ES BPSW):

$$time perl -Mntheory=:all -E 'is_prob_prime(divint(powint(10,$$_+1)-9*\$_-10,81)) and say for 1..6000;'
2497
3301
real    0m44.473s


PFGW (test is base 3 Fermat):

$$echo -e "ABC2 (10^(\a+1)-9*\a-10)/81\na: from 1 to 6000" > repsum.txt$$ time ./pfgw64 -k -u0 -f50 repsum.txt
(10^(2497+1)-9*2497-10)/81 is 3-PRP! (0.0793s+0.0362s)
(10^(3301+1)-9*3301-10)/81 is 3-PRP! (0.1333s+0.0615s)
real    0m42.963s


As the size gets larger, PFGW will get much faster relative to the others. E.g. times on Macbook Pro (2015) for checking n from 6001 to 7000 the times are 88.9s for Pari/GP, 26.0s for Perl/ntheory, and 13.4s for PFGW. Albeit for any discovered values we'd want to run another more strict test such as BPSW.