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 $p=2q-1$.
$\endgroup$ 21 Answer
$\begingroup$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-Za-z0-9_#@^&*()|§¥¡¢£¤©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)
