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 :D
Edit: Please find primes with position numbers $>9$.
$2,3,..$ are far too trivial.
3 Answers
$\begingroup$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
endThe proof of the "Sheldon Conjecture" was published a few months ago in the February edition of the American Mathematical Monthly.
$\endgroup$ 2 $\begingroup$I created a [script] you can play with here to test this out. Note that the answer depends on your numerical base -- among all bases I've tried, 10 seems to be the only base in which there's a Sheldon Cooper prime.
Base 16 seems promising, however -- it has a large number of "special emirps", and actually provides primes with the appropriate product of digits, which very few bases provide.
Can someone try base = 16, convbase = 2 (and perhaps other bases in multiple tabs) with a large uppercap (e.g. 10,000,000) using fastcount = false? It would take ~15 hours for an upper cap of 10 million -- or just 90 minutes for an uppercap of 1 million -- but I can't leave my laptop on for so long (the fan is malfunctioning).
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