73 - The Prime

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."

$$73_{10}=1001001_2$$

Leonard: "73 is the Chuck NorrisChuck Norris of numbers!"

Sheldon: "Chuck Norris wishes... all Chuck Norris backwards gets you is Sirron Kcuhc!"

Seventy-three is the 21st prime number. The previous is seventy-one, with which it comprises the 8th twin prime. It is also a permutable prime with thirty-seven. 73 is a star number.

73 is the largest minimal Primitive root in the first 100,000 primes. In other words, if p is one of the first 100,000 primes, then at least one of the primes 3, 5, 7, 11, 13, 17, ..., 73 is a primitive root modulo p.

73 is the smallest prime congruent to 1 modulo 24.

73 is an emirp, meaning that the reverse of 73, that is, 37, is also a prime number. 73 is also the 21st prime number while 37 is the 12th prime number.

The number 21 has prime factors 7 and 3. The number 21 in binary is 10101; 7 in binary is 111, 3 in binary is 11, and seventy-three in binary is 1001001. All of these are palindromes. In addition, of the 7 binary digits representing 73, there are 3 ones.

$$37 + 12 = 49 (7^2) \:\:and,$$ $$73 + 21 = 94 = 47 \cdot 2, \:\:\:47 + 2 = 7^2$$

Additionally, both 73 and its mirror, 37, are sexy primes twice over, as 31, 43, 67 and 79 are all prime numbers.

Every positive integer can be written as the sum of 73 or fewer sixth powers (see Waring's problem).

In base 5, the smallest prime with a composite sum of digits is 73.

• 73 is the only prime repunit in base 8 (1118).

• 73 is the smallest factor of the first composite generalized Fermat number in base 10 $ 10 ^ 4 + 1 = 10001 = 73\cdot137$ .