# Narcissus prime

This morning Futility Closet posted the following.

Repeat the string 1808010808 1560 times, and tack on a 1 the end. The resulting 15601-digit number is prime, and because it’s a palindrome made up of the digits 1, 8, and 0, it remains prime when read backward, upside down, or in a mirror.

I used Mathematica to verify that the number described above is indeed prime.

`PrimeQ[ 10*Sum[1808010808*10^(10 i), {i, 0, 1559}] + 1 ]`

After a little over two minutes, the function returned `True`.

## 12 thoughts on “Narcissus prime”

1. Mathematica’s PrimeQ function might not be right. It isn’t always true. Though it usually is.

2. The PrimeQ documentation doesn’t say whether it uses a probabilistic primality test, but given the size of the argument I thought it might.

3. John Venier

My very first real math proof was that any palidromic prime must have an odd number of digits, which this one does, so there’s that for what it’s worth.

4. John Venier

^ Except 11 of course, provided it is prime in the base in question.

5. Mark Spencer

@John Venier: Maybe you should have humbly said palindromes of even length are multiple of 11. And if by real math proof you mean 100..001 = 99..990 + 11, well… But hey, yeah, that’s nice, it works in any base, provided 9 is 10-1.

6. Palindrome primes I had heard of. But “upside down or in a mirror” is a new one. So the digits of this prime are invariant under right-left reading and also under the Klein four-group Z2xZ2. Who knew? [Assuming, of course, that “1” is written as “|”]

7. I don’t have access to Mathematica at home, but interestingly using WolframAlpha with your expression produces False.

8. You can’t rely on alpha, it usually kills the process that would take more than a few seconds. Nevertheless, it is an amazing engine.

9. John Venier

@Mark Spencer: Well, it was just a fun memory. Sheesh. Not all of us start out like Erdos. Present company excepted, of course.

10. David Harris

This looks cute, but there is nothing special about this number — it is basically random chance. By the Prime Number Theorem, the proportion of n-bit numbers which are prime is \$~1/n\$. Informally, if you generate “randomish” numbers with n bits there is a very high probability you will end up with a prime.

11. This came along at just the right time. My year 7’s have just been doing prime numbers so being able to discuss this particular prime was a bit of fun.

12. is prime 10^ 9000 000 000+3. false. try factor 1580 187 223. oeis. 2000. sci.math groups.google.com