# 223

This number is a prime. 223 is the least positive integer requiring maximum number of terms when expressed as a sum of positive 5-th powers. What is the maximum number of terms? Hint: It's an emirp. The number of primes and the number of composites that cannot be written as the sum of two primes, up to 223, are equal. [Honaker] The sums of the nth powers of its digits are prime for all n between 1 and 6 inclusive: sum of digits = 7, sum of squares of digits = 17, sum of cubes of digits = 43, sum of fourth powers = 113, sum of fifth powers = 307 and sum of sixth powers = 857. [Trotter] One Saros cycle is almost exactly 223 synodic months. Prime in the decimal expansion of square root of 5. [Haga] A chicken and human have 223 enzymes of identical sequence length. [Jolly] The prime factors of 2p - 1 are all of the form 2kp + 1, where k is a positive integer, and p is an odd prime. Fermat used this fact to show that 223 divides the Mersenne number M(37). A prime extracted from the names of the most famous Star Wars droids, i.e., R2-D2 and C-3PO. May the Force of Prime Numbers be with you! [Capelle] The sum of the digits of first primes up to 223 is 449. Note the prime digits and their squares. [Silva] The smallest prime showing a repeated prime digit. [Silva] The smallest prime whose reversal has more than two prime divisors. [Silva] The largest gear in the Antikythera mechanism most likely had 223 teeth in connection with the prediction of lunar eclipses. [Beedassy] The smallest prime formed from three prime digits. [Silva] 223 = -01+23+45+67+89. [Silva] The smallest prime that has more primitive roots below p/2 than above p/2. [Gudipati] "The children of Hashum, two hundred twenty and three." (Ezra 2:19, KJV) [Dorton] (223, 227, 229, 233) is the first tetrad of successive primes whose the digits of each prime are complementary of the digits of another tetrad of successive primes, i.e., (887, 883, 881, 877). [Loungrides] The only 3-digit isolated prime concatenated from two isolated primes, 2 and 23. [Loungrides] ß was encoded by ECMA at position 223 (hexadecimal DF). [Hess] 223 is the first of 13 consecutive primes squared having squares as the last three digits. [Bergot] The first Carol prime, i.e., a prime of form (2^x-1)^2-2, for x a composite, (x=4). Note that 223 is a prime-digit prime. [Loungrides] The starting prime of a sequence of five primes following this rule: The Kth prime is obtained from the previous one substituting each digit d by d^K: 223, 449, 6464729, 129625612962562401166561, 1325904977763231257776132590497776323125777632102401177767776312577761. [Rivera] The sum of the first 223 distinct-digit numbers is divisible by the 223rd distinct-digit number. [Gaydos] The expression 2^x + 2^x + 3^x is prime for x = 0 thru 6. [Anitha] German astronomer Johannes Kepler composed 223 footnotes to his novel Somnium—several times longer than the actual text. [Edward]

(There are 14 curios for this number that have not yet been approved by an editor.)