# Sheldon Cooper Primes

On the $73^{\text{rd}}$ episode of the Big Bang Theory, Dr. Sheldon Cooper, an astrophysicist portrayed by Jim Parsons $(1973 - \stackrel{\text{hopefully}}{2073})$ revealed his favorite number to be the sexy prime $73$

Sheldon :
The best number is $73$.
Why?
$73$ is the $21^{\text{st}}$ prime number.
Its mirror, $37$, is the $12^{\text{th}}$
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.

Up to 10,000,000 $\;\;$ (currently running until 100,000,000)

• Emirp with added mirror properties (as defined above):

• $+$ Mirror different from original prime:

• $+$ Binary representation of the prime is a palindrome:

• $+$ A concatenation of the factors of the position number of the prime yields the prime:

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