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.

Solutions Collecting From Web of "Sheldon Cooper Primes"

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

  • Emirp with added mirror properties (as defined above): $$2, \;\;\; 3, \;\;\; 5, \;\;\; 7, \;\;\; 11, \;\;\; 37, \;\;\; \text{and}\;\;\; 73.$$

  • $+$ Mirror different from original prime:$$37, \;\;\; \text{and}\;\;\; 73.$$

  • $+$ 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