Problems & Puzzles: Puzzles

Puzzle 586. 73 is the best prime

Perhaps you already know the popular Dr. Sheldon Cooper's claim:

"The best number is 73. Why? 73 is the 21st prime number. Its mirror (37) is the 12th and its mirror (21)

In our terms:

If n=12, p(r(n))=r(p(n))........(1)

Q. Can you get a larger solution for (1) or prove that there are no more solutions for n>12?

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 Note1: non-elegant solutions come when n & p(n) are both palindromes. Here we do not consider these solutions that -BTW- are the matter of our old Puzzle 51 that remains stuck since 2003!!!

Note2: I should thank to my friend Jaime Ayala for let me know about this popular claim.
 

Contribution came from Emmanuel Vantieghem

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Emmanuel wrote:

I think there might be other 'elegant' primes like  37, but then these will be greater than  9196155337

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Akash Betrieve wrote on March 12, 2020

A solution to Puzzle no. 586 can be found in a recent paper by Carl Pomerance, an esteemed number theorist. In brief, the paper outlines a proof that 73 is the only number that has the properties mentioned in Puzzle 586, "The Sheldon Conjecture". Two links to the paper will follow below. The proof is mainly based on the prime number theorem.
 

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