# 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 + 2^{3} * 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 = 2^{2} + 2^{2} = 2^{3}). [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 2^{q} - 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]

(10^{8} - 8)/8 is prime. [Luhn]

(10^{8+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) = (10^{n} - 1)/9 and T_{n} = the nth triangular number,
then P(n) = T_{R(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]