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!”
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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 ðŸ˜€
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): $$2, \;\;\; 3, \;\;\; 5, \;\;\; 7, \;\;\; 11, \;\;\; 37, \;\;\; \text{and}\;\;\; 73.$$
$+$ 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