Is (10^y - 1) x (10^y) + 1 prime for any value of y besides 2, 4, 6, and 8?

Chuck Seggelin seqfan at plastereddragon.com
Mon Nov 29 07:33:22 CET 2004

```Hello seqfans...

I was factoring some large numbers this weekend, and I noticed a few primes
that had very similar forms: 9901, 99990001, 999999000001, and
9999999900000001.  Essentially, these primes are the values produced by the
formula (10^y - 1) x (10^y) + 1 for the following values of y: 2, 4, 6, and
8.  I've noted these primes before and thought them interesting.

I was curious to know if there are any other values of y for which (10^y -
1) x (10^y) + 1 is prime.  I have since tested all values of y up to 550 and
have found no further primes of this form.  I am starting to wonder if there
are no others.  I find it odd that it works for all the even numbers between
1 and 9 and then nothing all the way up to 550.  Alas, I am just a hobbyist,
and if there are no other working values of y, I am at a loss to explain why
that would be.

If there are no other primes of this form, I'm sure a number theorist or
accomplished mathematician would be able to explain why.  Can any experts
out there want tell me if there is an obvious reason why (10^y - 1) x (10^y)
+ 1 could only produce primes when y is 2, 4, 6, or 8?  Or is it possible
(likely even) that (10^y - 1) x (10^y) + 1 will produce further primes at
larger values of y that I haven't tested yet?

(I note at the site
http://primes.utm.edu/curios/page.php/9999999900000001.html the author says
all values of y up to 50,000 have been tested with no further primes found.
The implication is that there is not a simple rule which prevents larger
values of y from producing primes.)