## Fascinating primes whose reversal is a prime

When you reverse the digits of a prime, you can either get…
- a composite number
- a prime

The reversal of a number abc… is …cba. A reversible prime is a prime that remains prime when its digits are reversed. Reversible primes can be divided into two sets: emirps and palindromic primes.

(1)   Palindromic primes: prime numbers whose decimal expansion is a palindrome
2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741, 15451, 15551, 16061, 16361, 16561, 16661, 17471, 17971, 18181, …

Table of n, a(n) for n=1 … 100197

(2)   Emirps (primes whose reversal is a different prime)
13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389, 701, 709, 733, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991, 1009, 1021, 1031, 1033, 1061, 1069, 1091, 1097, 1103, 1109, 1151, 1153, 1181, 1193, …

Table of n, a(n) for n=1 … 10000

73   is the   37-th   odd number
73   is the 21st prime number, and   37,   the 12th prime number.
The factors of 21 are:   1, 3, 7, 21
The prime factors of   21   are:   3 * 7
73   in binary : 1001001
It is a palindrome, it has 7 digits, and has 3 ones.
Equality using 3, 7, 37, and 73:     37*7 – 73*3 = (7 – 3)*(7 + 3)

3! * 7! – 37 = 30203    (a palindromic prime)

13   and   31
13   is the 1st emirp,   31,   the 3rd emirp

13^2 = 169   and its reversal   31^2 = 961.    (digits are reversed)

You can use the first 3 primes to express 13:   2^3 + 5 = 13
and the first two primes to express 31:   2^2 + 3^3 = 31

13*3 – 31*1 = (3 – 1)*(3 + 1)

79 & 97 :
Equality expressing   79,   97   and a prime number:    97,779,977 = 97 * 1,008,041
1,008,041 is the   79103-rd prime.

77 * 78 * 79 = 474,474 = 474 || 474   (palindromic number)

14141 = 179 * 79   (palindromic number)

79*9 – 97*7 = (9 – 7) * (9 + 7)

TO BE CONTINUED