## Random mathematics factoid.

March 4, 2011 by dberenstein

Well, sometimes I get five minutes off to goof around and I find silly facts, which as far as I can tell serve little to no purpose. Hence the name factoid is sometimes applied to them. The random novelty of the day is that the number

is prime. Moreover, it is a factor of . The only other prime factor is almost as pretty.

Beware of which surprise, surprise, it also happens to be prime. And so seems to be

If someone knows anything more about these primes, please let it be known. I found them by accident, but they seem to have been discovered before my time, so I can not name them after myself, nor get a patent for them.

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on March 4, 2011 at 2:03 amAaron BergmanThose are called repunits. People have spent a fair bit of effort looking for examples that are prime.

I can’t imagine you didn’t already work this out for yourself, but your factorization is the identity

(1/99)(100^n – 1) = (1/9)(10^n-1) * (1/11)(10^n+1)

for n odd.

on March 4, 2011 at 6:34 pmdberensteinHi Aaron:

This started with a silly e-mail my dad sent me so that after some arithmetic ‘magic’ one would get one’s age to be repeated many times in a row. Weird thing is that so many people in the internet think this is “amazing and beyond belief”. Obviously it was all because someone figured out how to factorize numbers like 101010101.

After monkeying around with those numbers the large primes built of ones started to pop up. I didn’t want to spend too much time doing research on them. This is why a blog is a perfect place for someone else to do the digging.

on March 4, 2011 at 2:40 amNamelessThese are two of the only four prime numbers of the form (10^x-1)/9 for x<1000.

The other two are 11 and (10^317-1)/9.

It is quite easy to show that sequences of 1's are composite as long as their lengths are composite.

on March 4, 2011 at 9:29 amLuboš MotlVerified by Wolfram, they’re primes.

Every kid from the kindergarten may easily see that the sequences of digits “1” are primes if their length belongs to the following set:

2,19,23,31,317,1031, …

The number 11….11 (which has 1,031 digits in total) is known as the Bolun (Boring Lumo Number). ;-)

More seriously, I calculated those primes and then made a Google searches which was the easiest way to find out that they’re called repunits:

http://en.wikipedia.org/wiki/Repunit

on March 4, 2011 at 8:44 pmNamelessOne more thought.

Simple heuristics based on the prime number theorem tells us that the number of such primes is infinite.

More specifically, I can estimate that, for x<N, the number of primes is (33/8 ln 10) ln ln N. But that estimate seems too conservative. Can you do better?

on March 14, 2011 at 3:14 amJerry BI looked at your 2 numbers made only with 1’s. Being a CS person, I looked to see if either numbers, seen as a binary number, would be prime.

Number #1: in decimal, 524,287, is a prime

Number #2: in decimal, 67108863, is not a prime with the factors of 3, 2731, 8191.