# 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 Fools prime." (April Fools' Day is an annual custom on 1 April consisting of practical jokes and hoaxes. Jokesters often expose their actions by shouting "April Fools!" at the recipient.)

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.

The conjecture that all odd integers greater than 9 are the sum of three odd primes is called the "weak" Goldbach conjecture. It was finally proved to be true by mathematician Harald Hefgott of the École Normale Supérieure in Paris.

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]

Among the first billion prime numbers, a prime ending in 9 is about sixty-five percent more likely to be followed by a prime ending in one than it is to be followed by a prime ending in nine.