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GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) 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] Lefttruncatable 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 (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 p1 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] ((0)! + (1)! + ... + (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) > (n) + sigma(n), so 8 is the only number n such that prime(n) = (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. 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 righttruncatable 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 ... . The smallest prime raised to the third power ('TWO CUBED') has eight letters. [Homewood] 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 primeprime index pairs. [Gaydos]
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