# Limerick primes

The other day, Futility Closet posted this observation:

10102323454577 is the smallest 14-digit prime number that follows the rhyme scheme of a Shakespearean sonnet (ababcdcdefefgg).

I posted this on AlgebraFact and got a lot of responses. One was from Matt Parker who replied that 11551 was the smallest prime with a limerick rhyme scheme.

So how many limerick primes are there? Well, there aren’t many candidates. A limerick prime has to have the form AABBA where A is an odd digit and B is any digit other than A. So for each of five choices of A, there are nine possible B’s. Here’s a Mathematica program to do a brute force search for limerick primes.

```    For[ j = 0, j < 5, j++,
For[ k = 0, k < 10, k++,
x = (2 j + 1)*11001 + 110 k;
If[ PrimeQ[x], Print[x] ]]]
```

It turns out there are eight limerick primes:

• 11551
• 33113
• 33223
• 33773
• 77447
• 77557
• 99119
• 99559

See the next post for Mathematica code to list all sonnet primes.

Update: See Lawrence Kesteloot’s code for a different kind of Limerick prime, a number that sounds like limerick when read outloud.

## 10 thoughts on “Limerick primes”

1. Incidentally, 99559 are the number of syllables per line in a limerick.

2. John V.

Thats really cool and it takes me back to my youth …

One of my first forays into numerical computing was an implementation of the Sieve of Eratosthenes in graphics memory (for space) to ennumerate primes. I noticed that 16661 was prime and then limited the output to palindromic primes. I happened to notice that all of the palindromic primes except 11 had an odd number of digits. That inspired my first mathematical proof — that 11 is the only palindromic prime with an even number of digits.

3. What if you consider 2 digits for a and b like AABBA = 13 13 12 12 13?

4. Here are additional limerick primes for A and B < 20,

7710107, 7713137, 7719197, 9913139, 9916169, 9919199, 1111202011,13134413, 13138813, 17172217, 17177717, 1717202017, 19191119, 19192219, 19194419, 1919111119, 1919161619, 1919171719.

5. EastwoodDC

Wishing I had time today to compose an appropriate limerick in response. :-)

6. Andrew

Two quintillion, seventy-seven
Trillion, one billion,
Twenty-four million,
One thousand two hundred and seven.

7. Andrew’s prime above is 2077311001024001207 for those who, like me, might need to look up the meaning of quintillion or quadrillion.

8. A functional version:

Select[Flatten[Table[
(2 j + 1)*11001 + 110 k,
{j, 0, 4, 1},
{k, 0, 9, 1}]], PrimeQ] // TableForm