Can $7$ be the smallest prime factor of a repunit?

Question

Repunits are numbers whose digits are all $1$. In general, finding the full prime factorization of a repunit is nontrivial.

Sequence A067063 in the OEIS gives the smallest prime factor of repunits. There are no $7$'s (just the number $7$, not as a digit in a different prime) that I can see in the sequence.

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$7$ does divide many repunits, but always does when $3$ is also a factor, so $7$ never shows up as the smallest prime factor. The table of the first $508$ smallest prime factors of repunits shows that $7$ is not the smallest prime factor of any of those repunits.

My question is can $7$ ever be the smallest prime factor of a repunit?

I tried to find a repunit with $7$ as the smallest factor but searched very far and $3$ is always a factor whenever $7$ is.

I tried to prove that it can't be but I have no idea how.

Answer 1

Can $7$ be the smallest prime factor of a repunit?

No, it cannot.

The main idea is as follows: If the $n^\text{th}$ repunit $R(n)$ is divisible by $7$, then $n$ is divisible by $6$, which means $n$ is divisible by $3$, which means $R(n)$ is divisible by $3$, so $7$ cannot be the smallest prime factor.

The full argument is fleshed out below.

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Let $R(n)$ be the $n^\text{th}$ repunit, e.g. $R(4) = 1111$.

One interesting property about repunits is that if an integer $m$ divides $R(n)$, then $m$ divides $R(2n)$, and $m$ divides $R(3n)$, and so on. This is because, for any positive integer $n$, we have the factorization $$R(2n) = \left(10^{n}+1\right)R(n)$$ and, more generally, for any positive integers $k$ and $n$, we can factor as follows:

$$R(kn) = \left(\displaystyle\sum\limits_{j=0}^{k-1} 10^{jn}\right)R(n)$$

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What this means, for your problem, is if a prime $p$ divides a repunit $R(n)$, then it will also divide $R(kn)$ for all $k$.

This tells us that:

• Every other repunit is divisible by $11$. $$11 \,|\, R(2) \implies\,\Big(\,\,\,n \equiv 0 \pmod{2} \iff 11 \,|\, R(n) \,\,\,\Big)$$ Here, the "$\iff$" bidirectionality follows from the fact that $11 \not| \,\,R(1)$.

• Every third repunit is divisible by $3$. $$3 \,|\, R(3) \implies \,\Big(\,\,\,n \equiv 0 \pmod{3} \iff 3 \, |\, R(n) \,\,\,\Big)$$ Here, the "$\iff$" bidirectionality follows from the fact that $3 \not| \,\,R(1)$ and $3 \not| \,\,R(2)$.

• Every third repunit is divisible by $37$. $$37 \,|\, R(3) \implies \,\Big(\,\,\,n \equiv 0 \pmod{3} \iff 37 \, |\, R(n) \,\,\,\Big)$$ Here, the "$\iff$" bidirectionality follows from the fact that $37 \not| \,\,R(1)$ and $37 \not| \,\,R(2)$.

• Every sixth repunit is divisible by $13$.$$13 \,|\, R(6) \implies \,\Big(\,\,\,n \equiv 0 \pmod{6} \iff 13 \, |\, R(n) \,\,\,\Big)$$ Here, the fact that each of $1, 11, 111, 1111, 11111$ are not divisible by $13$ implies the "$\iff$" bidirectionality.

• Every sixth repunit is divisible by $7$.$$7 \,|\, R(6) \implies \,\Big(\,\,\,n \equiv 0 \pmod{6} \iff 7 \, |\, R(n) \,\,\,\Big)$$ Here, the fact that each of $1, 11, 111, 1111, 11111$ are not divisible by $7$ implies the "$\iff$" bidirectionality.

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This last fact implies that $7$ can never be the smallest prime factor of a repunit, because whenever $n \equiv 0 \pmod{6}$, it's also true that $n \equiv 0 \pmod{3}$, so whenever $7$ is a factor of a repunit, $3$ will be as well.

By a similar argument, $13$ and $37$ also cannot be the smallest prime factor of a repunit.

Answer 2

The repunits as a recurrence obey $$ r_{n+1} = 10 r_n + 1 $$ see https://en.wikipedia.org/wiki/Pisano_period

The code making the table of Pisano periods below. The simple outcome: the possible values of $r_n \pmod{21}$ are $$ 0, 1, 2, 6, 11, 19 . $$ In particular, the remainder after dividing by 21 can be $6,$ but is never $7$ or $14 \pmod {21}.$ Thus, $r_n$ can be divisible by $3$ alone, without any $7.$ Once $r_n$ is divisible by $7,$ the only possibility is $0 \pmod {21},$ meaning also divisible by $3$

   mpz_class rep = 0;
      for(int n = 1; n <= 45; ++n){
         rep = rep * 10 + 1;
      cout << "  n: " << n  << "  rep: " << rep << " mod 21: " << rep % 21 << endl;
       }

with no extra symbols. This is also a complete proof:

jagy@phobeusjunior:~$ ./mse

Wed Nov 27 18:45:37 PST 2019

  n: 1  rep: 1 mod 21: 1
  n: 2  rep: 11 mod 21: 11
  n: 3  rep: 111 mod 21: 6
  n: 4  rep: 1111 mod 21: 19
  n: 5  rep: 11111 mod 21: 2
  n: 6  rep: 111111 mod 21: 0
  n: 7  rep: 1111111 mod 21: 1
  n: 8  rep: 11111111 mod 21: 11
  n: 9  rep: 111111111 mod 21: 6
  n: 10  rep: 1111111111 mod 21: 19
  n: 11  rep: 11111111111 mod 21: 2
  n: 12  rep: 111111111111 mod 21: 0
  n: 13  rep: 1111111111111 mod 21: 1
  n: 14  rep: 11111111111111 mod 21: 11
  n: 15  rep: 111111111111111 mod 21: 6
  n: 16  rep: 1111111111111111 mod 21: 19
  n: 17  rep: 11111111111111111 mod 21: 2
  n: 18  rep: 111111111111111111 mod 21: 0
  n: 19  rep: 1111111111111111111 mod 21: 1
  n: 20  rep: 11111111111111111111 mod 21: 11
  n: 21  rep: 111111111111111111111 mod 21: 6
  n: 22  rep: 1111111111111111111111 mod 21: 19
  n: 23  rep: 11111111111111111111111 mod 21: 2
  n: 24  rep: 111111111111111111111111 mod 21: 0
  n: 25  rep: 1111111111111111111111111 mod 21: 1
  n: 26  rep: 11111111111111111111111111 mod 21: 11
  n: 27  rep: 111111111111111111111111111 mod 21: 6
  n: 28  rep: 1111111111111111111111111111 mod 21: 19
  n: 29  rep: 11111111111111111111111111111 mod 21: 2
  n: 30  rep: 111111111111111111111111111111 mod 21: 0
  n: 31  rep: 1111111111111111111111111111111 mod 21: 1
  n: 32  rep: 11111111111111111111111111111111 mod 21: 11
  n: 33  rep: 111111111111111111111111111111111 mod 21: 6
  n: 34  rep: 1111111111111111111111111111111111 mod 21: 19
  n: 35  rep: 11111111111111111111111111111111111 mod 21: 2
  n: 36  rep: 111111111111111111111111111111111111 mod 21: 0
  n: 37  rep: 1111111111111111111111111111111111111 mod 21: 1
  n: 38  rep: 11111111111111111111111111111111111111 mod 21: 11
  n: 39  rep: 111111111111111111111111111111111111111 mod 21: 6
  n: 40  rep: 1111111111111111111111111111111111111111 mod 21: 19
  n: 41  rep: 11111111111111111111111111111111111111111 mod 21: 2
  n: 42  rep: 111111111111111111111111111111111111111111 mod 21: 0
  n: 43  rep: 1111111111111111111111111111111111111111111 mod 21: 1
  n: 44  rep: 11111111111111111111111111111111111111111111 mod 21: 11
  n: 45  rep: 111111111111111111111111111111111111111111111 mod 21: 6

Wed Nov 27 18:45:37 PST 2019

jagy@phobeusjunior:~$ 

Answer 3

$7|1001 = 7*11*13$

So $7|1001*111 = 111,111$ so if $7|\underbrace{1111.....1}_n$ then

$7|\underbrace{11111.....1}_n - 111111$

$ = \underbrace{1111111....111}_{n-6}000000$

and as $10$ and $7$ are relatively prime. $7|\underbrace{1111111....111}_{n-6}$

So by induction if $n \equiv a \pmod 6$ then $7|\underbrace{1111...1}_{a}$.

So we just need to show when does $7|1,11,111,1111,11111,111111$

And that is only $7|111111 = 111*1001 = (3*37)*(7*11*13)$.[*]

So $7|\underbrace{1111......1}_n$ if and only if $6|n$ in which case $3,11,13,37$ will also divide the repunit.

Thanks to Zubin Mukerjee for the hint.

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[*]. $7|\underbrace{111...1}_{a < 6}\iff 7|\underbrace{111...1}_{6-a}$ but $7\not \mid 1(a=1)$ and $7\not \mid 11 (a=2)$ and so $7\not \mid 11111 (6-1)$ and $7\not\mid 1111 (6-2)$ while $7\not \mid 111 =3*37$