# 8

This number is a composite. Every odd perfect number has at least 8 distinct prime factors. [Hagis] 8 times the 8th prime has sum of digits equal to 8. A prime quadruple is four consecutive primes, such that the first and last differ by 8. - 1 + 23 * 45678 is prime. [Kulsha] The smallest number which is the sum of two distinct odd primes. The rightmost nonzero digit in 4013! (4013 is prime!) is 8. [Ottens] The smallest cube which is the sum of a twin prime pair (3 + 5). [Trotter] The smallest number which is both a sum of prime squares and a prime cube (8 = 22 + 22 = 23). [Kulsha] The absolute difference between two odd prime squares is always a multiple of 8. [Capelle] A Gaussian prime is a Gaussian integer p with exactly 8 divisors. [Smith] Left-truncatable primes p of length n with the additional property that no prime with length n + 1 can have its leftmost digit removed to produce p are called Henry VIII primes. 8 ones plus 8 is prime. [Opao] 8 is the smallest sum of two factorials of distinct primes (2! + 3!). [Gevisier] The largest number in which n is exactly twice π(n). [Murthy] Let p and q be odd primes. If p divides 2q - 1, then p ≡ 1 (mod q) and p ≡ ± 1 (mod 8). The largest composite number such that all its proper divisors + 1 are primes. [Murthy] (108 - 8)/8 is prime. [Luhn] (108+8 - 8)/8 is also prime. Thanks Mr. Luhn! [Poo Sung] The first 'not possible' occurrence of summing k consecutive primes such that the total is prime happens when k = 8. [De Geest] No Fibonacci number greater than 8 is ever of the form p-1 or p+1 where p is a prime number. [Honsberger] The smallest composite Fibonacci number. [Gupta] The 8th Fibonacci number plus and minus 8 is prime, i.e., F(8)-8 and F(8)+8 are primes. It is the smallest Fibonacci number to have this property. [Opao] The product of the first k nonzero Fibonacci numbers + 1 is prime for k = 1, 2, 3, 4, 5, 6, 7, and 8. Eight eights raised to the eighth power plus one is prime. I.e., 88888888^8 + 1 is prime. [Opao] The first difference between consecutive primes of 8 is after 8*11+1. This is the last maximal gap of 4 consecutive even numbers. [Nicholson] (8^8+88888)/8 is prime. Note that eight 8s are used. [Firoozbakht] (π(0)! + π(1)! + ... + π(k)!) is prime only for k = 1,2,..,8 (composite for k>8). [Firoozbakht] 8 is the smallest composite number which can be represented as sum of two primes (i.e., 3 + 5) as well as sum of two composite numbers (i.e., 4 + 4). [Capelle] Let R(n) = (10n - 1)/9 and Tn = the nth triangular number, then P(n) = TR(n) + 1 generates primes for n = 1 through 8. [Wesolowski] The only composite digit that is not a semiprime. [Silva] For n > 8, prime(n) > π(n) + sigma(n), so 8 is the only number n such that prime(n) = π(n) + sigma(n). [Firoozbakht] The number of autobiographical primes is composite, but the number of autobiographical composites is prime. [Capelle] The only known n such that the number of nonalternating knots with n crossings is prime. When n=8, there are 3 such knots. [Post] The Riemann Hypothesis was number 8 on the list of problems that David Hilbert believed should set the course for the mathematical explorers of the twentieth century. prime(8) = sigma(8) + phi(8). It is the largest number with this property. [Firoozbakht] 8 is the smallest cube that is the sum of two consecutive Sophie Germain primes (3 + 5). Can you find the next cube with this property? [Gaydos] The second smallest Fibonacci number equal to the sum of two consecutive Fibonacci primes: 8 = 3 + 5. [Rivera] The only known cube that is the sum of the first n odd primes (3+5=8). [Bajpai] The only integer that can be expressed as the average of first n consecutive right-truncatable primes, (n=5), i.e., (2+3+5+7+23)/5. [Loungrides] The digit 8 has yet to make an appearance in the constant 0.269606351971674 ... . [Honaker] 'TWO CUBED' has 8 letters. [Homewood] The digital root of product of twin primes, except (3, 5). [Gupta] 8 is the last digit to appear in both a prime number and its prime index, first appearing in 787 and 138. The other 9 digits all first appear in smaller prime-prime index pairs. [Gaydos] The sum of the first 8 composite numbers and the sum of the first 8 prime numbers differ only by 1. Does the sum of the first n composite numbers ever divide the sum of the first n primes? [Honaker] The smallest cube k such that 3^k + k and 3^k - k both are prime. [Bajpai] 8 = 2^3 is the only possible cube "centered and sandwiched" in the middle of two pairs of twins (3,5) and (11,13). That is to say 2^3=8=(3+13)/2=(5+11)/2. [Rivera]

(There are 10 curios for this number that have not yet been approved by an editor.)