How to prove 1111......11 (91 digits) is a prime or composite number?

Question

How to prove $1111......11$ ($91$ digits) is a prime or composite number?

My Approach:

$1111......11$ can be expressed as $10^{0}+10^{1}+10^{2}+...…..+10^{90}$

Using summation of a geometric progression formula,

$$10^{0}+10^{1}+10^{2}+...…..+10^{90} = 1\frac{(10^{91}-1)}{(10-1)}=\frac{(10^{91}-1)}{(9)}$$

I do not know how to proceed after this step. Kindly guide me how to solve this problem.

Suggest google on "repunits" or "repunit primes". – coffeemath Aug 20 '19 at 06:49 — coffeemath, Aug 20 '19 at 06:49

score 5 · Answer 1 · answered Aug 20 '19 at 06:51

5

Let me illustrate on a smaller repunit:

$111111=11\cdot10101$ and $111111=111\cdot1001$, because $6=3\cdot2$

Yours is the same problem, except $91=7\cdot13$.

answered Aug 20 '19 at 06:51

score 5 · Accepted Answer · answered Aug 20 '19 at 06:53

5

If $m=pq$ where $p>1,q>1$ are natural numbers

$$\dfrac{a^m-1}{a-1}=\dfrac{a^{pq}-1}{a-1}=\dfrac{(a^p)^q-1}{a-1}$$ will be divisible by $\dfrac{a^p-1}{a-1}$ which will be $>1$ as $p>1$

answered Aug 20 '19 at 06:53

lab bhattacharjee

Thank you for your answer. So the number $111...11$ (n times) is prime if and only if n is prime. Am I right? – Vishnu Aug 20 '19 at 06:57
3

@Intellex, If and only if is not correct. This is only necessary condition, not sufficient – lab bhattacharjee Aug 20 '19 at 07:01
@Intellex, See https://en.wikipedia.org/wiki/Repunit#Decimal_repunit_primes – lab bhattacharjee Aug 20 '19 at 07:02
Thank you for your answer and link. May be 111 could be an example, where the number of 1's is 3, i.e., prime but is divisible by 3. – Vishnu Aug 20 '19 at 08:40

Mohammad Riazi-Kermani · Answer 3 · 2019-08-20T11:15:50.330

0

Note that$$91=13(7)$$ so you may factor $$1111111$$ and your number is $$1111111(10^{12\times {7}} + 10^{11\times 7}+10^{10\times 7}+...1)$$

So it is composite.

edited Aug 20 '19 at 11:15

answered Aug 20 '19 at 07:12

Did you mean $12\times\color{red}7$? – J. W. Tanner Aug 20 '19 at 10:01
1

@J.W.Tanner Thanks for the comment. I fixed my error. – Mohammad Riazi-Kermani Aug 20 '19 at 11:17

3 Answers3