# 9

This number is a composite. If the digit sum of n!, S(n!), is the product of 9 and any prime larger than n, then S(n!) cannot divide n!. All digits of the prime 2 * 103020 - 1 are 9 except 1. It contains 3021 digits. [Williams] The sum of the first 9 consecutive prime numbers is a perfect square. [Honaker] If odd perfect numbers exist, they are of the forms 12n + 1 ... or 36n + 9. [Touchard] 9 is the smallest "April Fool prime." The sequence begins 9, 21, 27, 33, 39, 49, 51, 57, 63, 69, 77, 81, 87, 91, 93, 99, 111, ... . For every prime p with p not equal to 2 and p not equal to 5, there is some number with all digits equal to 9 such that p divides evenly into this number. Goldbach conjectured that every odd integer greater than or equal to 9 can be represented as the sum of three odd primes. There are no consecutive-digit primes starting with 9 with digits in descending order. [Madachy] Define a certain number of irregularly marked points, n, along the rim of a paper circle, then cut along straight lines that join all possible pairs of points. If n = 9, a prime number of separate pieces will be created. 163 to be exact! The smallest odd Giuga number must have at least 9 prime factors. If a is greater than b, and b is greater than or equal to 1, then an + bn has a primitive prime factor with the exception of 23 + 13 = 9. 9 times the 9th prime has a sum of digits equal to 9. There are no clusters (groups) of 9 twin prime pairs less than 1014. [DeVries] Washington University in St. Louis provides a page that calculates the prime factors of a number (with a maximum of 9 digits). There are at least 9 prime numbers between x3 and (x + 1)3 for x greater than or equal to π, assuming the Riemann Hypothesis is true. The smallest odd composite number. [Gupta] 109 + 9 is prime. [Gupta] Two raised to the 9th power plus and minus 9 are primes! [Hoefakker] The first digit to appear as an end-digit in two consecutive primes (139 and 149). [Silva] 19, 109, 1009 and 10009 are primes. No other digit can replace the 9 and yield four primes. [Friend] The number of known positive integers which are the sum of two primes in exactly two ways is a prime square. [Capelle] 2^^n-9 = 2^(2^(2^(....(2^2)...)))-9 is (for large enough n) always divisible by both 7 and 11. Note that 9 is midway between 7 and 11. [Hartley] There are exactly 3=(sqrt(9)) pandigital improper fractions that reduce to 9 (provided each digit is used once). [Patterson] 9 is the only number m such that m = π(π(m)!). [Firoozbakht] The 9th Fibonacci number plus 9 is prime. [Losnak] The only composite digit that can appear as end-digit of a prime. [Silva] The only non-prime digit that is the difference of consecutive squares. [Silva] 10*(22n + 1) + 9 gives primes for n = 1 to 7. Therefore, there are 7 known Fermat numbers which yields primes when a 9 is appended. [Wesolowski] The smallest composite number n such that both 2n+n and 2n-n are prime. That is, 29+9 = 521 and 29-9 = 503 are prime. [Poo Sung] Three more semiprimes can be consecutively formed from 9 by iterating the process described in A227942. The smallest semiprime such that all permutations of concatenations with its factors are semiprimes. [Sariyar] The only known integer n, such that 2^n-n^2 and 2^n+n^2 are both primes, i.e., 431 and 593. [Loungrides] Regular nonagon is the only regular polygon such that n^n +(n-2)*180/n (387420629) and ((n-2)*180/n)^n+n (20661046784000000009) are both prime. Note that (n-2)*180/n is an internal angel of a regular polygon. [Sariyar] Smallest composite number n whose number of circular loops equals omega(n), i.e., the number of distinct primes dividing n. Circular loops occur in the digits 0, 6, 8, or 9 only. Note that the digit 8 contains two loops. The sequence begins 9, 16, 18, 28, ... . [Honaker] The composite digit which first appears in a prime. [Silva] The smallest composite number whose both concatenations with its home prime, (i.e., 311), in order and reverse order, 9311 and 3119 are primes. [Loungrides] Only 9 numbers have been found that are the sums of two distinct primes in exactly two ways. [Sariyar]