Prime number?

Number Theory Level 3

It is known that 11 is a prime. Does any prime numbers of the form $\underbrace{111\ldots 1}_{n \,\text{times}}$ , where $n>2$ , exist?

No. Yes

1 solution

Chan Lye Lee
Nov 8, 2018

Note that $\underbrace{11\ldots 1}_{19 \,\text{times}} = \sum_{k=0}^{18}10^k$ is a prime.

There is a conjecture that there are infinite many primes of the form $\underbrace{11\ldots 1}_{n \,\text{times}}$ .

Next prime is $\underbrace{111\cdots 11}_{\text{23 times }}$ .

Naren Bhandari - 2 years, 7 months ago

Values for $n$ such that those numbers are prime are

$2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343...$

but only the first five are proven to be primes. The others are only "probable primes".

A Former Brilliant Member - 2 years, 7 months ago

Any proof for existence?

Vishwash Kumar ΓΞΩ - 1 year, 8 months ago