# 163

This number is a prime.

A prime whose reversal is another prime (19) squared. [Trigg]

In the April 1975 issue of *Scientific American*, Martin Gardner wrote (jokingly) that Ramanujan's constant (*e*^(π*sqrt(163))) is an integer. The name "Ramanujan's constant" was actually coined by Simon Plouffe and derives from the above April Fool's joke played by Gardner. The French mathematician Charles Hermite (1822-1901) observed this property of 163 long before Ramanujan's work on these so-called "almost integers." [Aitken]

The largest Heegner number. [Croll]

163 = "is a prime number" by adding the letters in the alphabet code, i.e., a = 1, b = 2, c = 3, etc. [Necula]

163 is the smallest prime number that is a factor of more than one number of the form p# - 1 (163 divides both 67# - 1 *and* 79# - 1).

Let the cs(p) be the cumulative digit sum of all the primes 2 to p (e.g., cs(11)=2+3+5+7+1+1=19). There are the only four known primes such that cs(p)=2p; they are 5, 23, 47 and 163. [Vrba]

163 = 1+2*3^4. [Oliver-Lafont]

163 is the only known number *m* (up to 3*10^10) such that *m* + 4**n*^2 for *n* = 0, 1, 2, ... , 19 are prime. [Firoozbakht]

Conjectured to be the largest prime that can be represented uniquely as the sum of three squares (1^2 + 9^2 + 9^2). Note that squares are allowed to be zero. [Noe]

The smallest score impossible to make using up to but not more than 3 darts. [Geach]

The sum 37 + 59 + 67 of all 2-digit irregular primes. [Poo Sung]

The smallest prime *p* whose *p*th power *p*^{p} contains a pandigital substring: 163^{163} = 38599...(5941863207)...95547. Note that π(163) is a semiprime with larger prime factor 19 which is the smallest prime *q* whose *q*th power
*q*^{q} is pandigital, and that the concatenations 16319 and 19163 are also primes. [Beedassy]

According to Cam McLeman, 163 is the coolest number in existence.

163 is the least number k such that decimal representation of 1/k has period of length 81. It is one of a few exceptions to the rule that k=3^(n+2) is the least number with 1/k having period 3^n. [Noe]

The smallest 3-digit prime whose absolute value of the differences between any two of its digits are also prime. [Green]

Some Major League Baseball seasons consists of 163 games. [Post]

The smallest prime whose reversal is a Hoax number. [Loungrides]

163 and 631 are two primes consisting of the same distinct digits, (1, 3, 6), such that one of these digits is twice the product of the other two, i.e., 6 = 2*(1*3). Note that a concatenation of these primes creates a Cyclops prime, i.e., 6310163. [Loungrides]

Monument Valley is accessible from U.S. Highway 163.

The largest squarefree integer n such that the ring of integers of the field Q(sqrt(-n)) has unique factorization. [Luen]

The prime number 163 contains only the three positive triangular digits, as does the 163rd triangular number (13366). [Gaydos]

CD163 is a protein encoded by the CD163 gene in humans. [Ford]

Buying 163 lottery tickets in a 49 number lottery guarantees at least a 3-match. [Homewood]

November 2022 will mark 163 years since Bernhard Riemann posed a seminal question about the distribution of prime numbers.

The first 163 multiples of 6 are adjacent to a total of 163 prime numbers. [Gaydos]

There are 163 mathematical proofs of Legendre-Gauss law of quadratic reciprocity. [Olry]