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
