# First missing Curio!

We have presented prime curios for hundreds of integers, but still have missed so many! The first prime number which is missing a prime curio is

5659 | 5743 | 5783 | 5827 | 5843 |

5869 | 5927 | 5981 | 6067 | 6113 |

6197 | 6203 | 6229 | 6299 | 6311 |

Does that mean there is no prime number related curiosity about this integer?

No, just that we have not found one worthy of inclusion yet. In fact, below is a proof (okay, a joke proof), that every positive integer has an associated prime curio. So if you know a great curio for 5653, please submit it today!

First we need a definition. We will be a little stronger than Merriam-Webster's definition of curio and make our curios short:

A prime curio aboutnis a novel, rare or bizarre statement about primes involvingnthat can be typed using at most 100 keystrokes.

**Theorem:**Every positive integer

*n*has an associated prime curio.

**"Proof":** Let S be the set of positive integers for which there is no associated prime
curiosity. If S is empty, then we are done. So suppose, for proof by contradiction, that S is not
empty. By the well-ordering principle S has a least element, call it *n*. Then
** n is the least positive integer for which there is no associated prime curio**.
But our last statement is a prime curio for

*n*, a contradiction showing S does not have a least element and completing the proof.

(For further discussion of this pseudo-proof, see the page a Curious Paradox.)