On the 73rd episode of the Big Bang Theory, Dr. Sheldon Cooper, an astrophysicist portrayed by Jim Parsons (1973−hopefully2073) revealed his favorite number to be the sexy prime 73
Sheldon :
“The best number is 73.
Why?
73 is the 21st prime number.
Its mirror, 37, is the 12th
and its mirror, 21, is the product of multiplying 7 and 3
… and in binary 73 is a palindrome, 1001001, which backwards is 1001001.”Leonard : “73 is the Chuck Norris of numbers!”
Sheldon : “Chuck Norris wishes… all Chuck Norris backwards gets you is Sirron Kcuhc!”‘
My question is basically this: Are there any more Sheldon Cooper primes?
But how do I define a Sheldon Cooper Prime? Sheldon emphasizes three aspects of 73
It is an emirp with added mirror properties
(ie, the prime’s mirror is also a prime with position number mirrored)A concatenation of the factors of the position number of the prime yields the prime.
Binary representation of the prime is a palindrome
I think having all three properties exist simultaneously in a number is difficult.
So, a prime satisfying the first property is good enough.So, I define a Sheldon Cooper Prime as an emirp with added mirror properties.
Good Luck finding them 😀
Edit: Please find primes with position numbers >9.
2,3,.. are far too trivial.
Answer
Up to 10,000,000 (currently running until 100,000,000)

Emirp with added mirror properties (as defined above): 2,3,5,7,11,37,and73.

+ Mirror different from original prime:37,and73.

+ Binary representation of the prime is a palindrome: 73.

+ A concatenation of the factors of the position number of the prime yields the prime: 73.
Matlab Code
clc
clear
for i = 1:10000000
% Prime:
if (isprime(i))
cont = 1;
else
cont = 0;
end
% 1. It is an emirp with added mirror properties:
if (cont == 1)
mirror_i = str2double(fliplr(num2str(i)));
if (isprime(mirror_i))
cont = 1;
else
cont = 0;
end
end
if (cont == 1)
p_i = length(primes(i));
p_mi = length(primes(mirror_i));
mirror_p_i = str2double(fliplr(num2str(p_i)));
if (mirror_p_i == p_mi)
cont = 1;
disp(' ')
disp(' ')
disp(['>> ',num2str(i)])
disp(['Satisfies Condition 1: ',num2str([mirror_i,p_i,p_mi])])
else
cont = 0;
end
end
% 2. Mirror different from original prime:
if (cont == 1)
if (i == mirror_i)
cont = 0;
else
cont = 1;
disp('Satisfies Condition 2')
end
end
% 3. Binary representation of the prime is a palindrome:
if (cont == 1)
bin = dec2bin(i);
mirror_bin = fliplr(num2str(bin));
if (bin == mirror_bin)
cont = 1;
disp(['Satisfies Condition 3: ',num2str(str2double(bin))])
else
cont = 0;
end
end
% 4. A concatenation of the factors of the position number of the prime
% yields the prime:
if (cont == 1)
if (prod(sscanf(num2str(i),'%1d')) == p_i)
disp('Satisfies Condition 4')
end
end
end
Attribution
Source : Link , Question Author : Nick , Answer Author : mzp