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Same Digit Prime

Problem

11 is the smallest prime made up of the same digit.

Explain why a number made up of the same digit can only be prime if the repeating digit is one AND the number of digits is itself prime.


Solution

Clearly a number of the form $aaa...$ will be divisible by $a$; for example, 77777 is divisible by 7.

Hence such a number can only be prime if $a = 1$.

Consider the number 111111111111111, which is made up of fifteen ones. It should be clear that as $15/5 = 3$, it will be divisible by 111.

 
1
001001001001
111111111111111111

That is, 111111111111111/111 = 1001001001001.

In other words, if the number is made up of a string of $k$ ones, where $k$ = $ab$, then it will certainly be divisible by a string of $a$ (or $b$) ones.

Hence the only numbers made up of the same digit that are prime must be made up of a prime number of ones.

It is interesting to note that there are only five known repunit primes; that is, primes made up of a string of 2, 19, 23, 317, and 1031 ones. However, in 1999 and 2000 respectively strings consisting of 49081 and 86453 ones were found to be probable primes, which means that although they are still not proven to be prime, they have passed primailty tests for sufficiently many non-trivial cases that it is highly probable that they are prime.

Problem ID: 256 (01 Jan 2006)     Difficulty: 2 Star

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