# 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

6359 | 6373 | 6389 | 6599 | 6607 |

6619 | 6679 | 6763 | 6791 | 6803 |

6827 | 6959 | 6967 | 6971 | 6997 |

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 6343, 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.)