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(another Prime Pages' Curiosity)
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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
4603 followed by:
46394643465146734703
47214733475948134817
49094933496749694987
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 4603, 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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