The book is now available! First missing Curio!
(another Prime Pages' Curiosity)
Prime Curios!
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The nth Prime Page will now find any of the first 2.623˙1015 primes or π(x) for x up to 1017.

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

4127 followed by:
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 4127, 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 about n is a novel, rare or bizarre statement about primes involving n that 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.)

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