In addition to being probably the only Sheldon Cooper prime, $73$ is a reversible prime $p$ (or emirp), such that its reverse is $q=(p+1)/2$. It is not hard to see that all other reversible primes with this property must have the form $799\dots993$, but is $73$ the only such prime?

In addition one can show that for a reversible prime pair $(p,q)$ relation $$p=kq\pm 1$$ for natural $k$ can only be possible for $p=2q-1$.

  • 1
    Up to 2000 9's, it is. – rogerl Jul 16 '15 at 21:03

I'm not a mathematician but was intrigued as to whether $37:73$ prime pair are unique. As there's been no answer I wanted to share my answer:

Within the first 25 million primes $37:73$ are unique when represented in base 10. However, there are a few other unique prime/emirp pairs in other bases as shown below

As a reversible number isn't a property of a number itself but rather a feature of how it is represented, I tried rebasing the first 25 million prime numbers using different radices to see if any 'prime/emirp pairs' existed among the different bases. I searched the first 25 million prime numbers with radices of 2-100 looking for pairs that additionally meet the following criteria (also properties of the $37:73$) to give a cleaner answer:

  • The ‘emirp’ prime isn’t a palindrome prime (eg 101)

  • The ‘nth’ value of the ‘prime/emirp pair’ should also be non-palindrome primes of each other (37 is the 12th prime and 73 is the 21st prime)

To rebase, I used the following characters in order to visualise any reversible primes:

  • A-Z
  • a-z
  • 0-9
  • _#@^&*()|§¥¡¢£¤©ª«¬®¯°±²³µ¶»¿ÆÞØßîÿÑ×

Applying this, I get a unique pair (within the first 25 million primes anyway) for each of the following bases (# represents the position in the prime sequence) - all other bases have no results:

      Base10-----------------------    |Base N-------------------------
Base   Prime (#)       Emirp (#)       |Prime (#)         Emirp (#)
 2        67 (19)         97 (25)      |BAAAABB (BAABB)   BBAAAAB (BBAAB)
 4      1627 (258)      3673 (513)     | BCBBCD (BAAAC)    DCBBCB (CAAAB)
 9      1163 (192)      1747 (272)     |   BFDC (CDD)        CDFB (DDC)
10        37 (12)         73 (21)      |     DH (BC)           HD (CB)
11     64633 (6465)   119233 (11235)   |  EEGBI (EJEI)      IBGEE (IEJE)
13    257353 (15090)  257353 (22626)   |  FKBAJ (GLDK)      JABKF (KDLG)
15     24443 (2714)    29077 (3162)    |   HDJI (MAO)        IJDH (OAM)
17       991 (167)      1567 (247)     |    DHF (JO)          FHD (OJ)
31    353603 (30257)  535133 (44207)   |   LadR (BAPB)       RdaL (BPAB)
78     32533 (3491)    44699 (4646)    |    FbH (s7)          HbF (7s)
89       523 (99)       6947 (891)     |     F© (BK)           ©F (KB)

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