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The first pair of consecutive primes that differ by 6 are 23 and 29. 6 is the smallest number which is the product of two distinct primes. [Brueggeman] The only mean between a pair of twin primes which is triangular. [Shanks] The probability that a number picked at random from the set of integers will have no repeated prime divisors is 6/^{2}. [Chartres] There exists at least one prime of the form 4k + 1 and at least one prime of the form 4k + 3 between n and 2n for all n greater than 6. [Erdös] Prime doubles of the form (p, p + 6) are called sexy primes. + 1 + 234 * 5^{6} are twin primes. [Kulsha] The product of the first n Mersenne prime exponents + 1 is prime for n = 1, 2, 3, 4, 5 and 6. If p + 1 is a twin prime greater than 6 then p = 6n, where n is a positive integer. [Goldstein] 6 is the smallest composite number whose sum of prime factors is prime. [Gupta] The smallest invertible brilliant number. [Capelle] 6 is the only number such that the sum of all the primes up to 6 equals the sum of all the composite numbers up to 6. [Murthy] 6*66*666*6666*66666*666666 + 1 is prime. [Poo Sung] The only perfect number that can be sandwiched between twin primes. [Gupta] 6 is the smallest composite number such that the concatenation (23) as well as the sum (5) of its prime factors are prime. [De Geest] It is a theorem that primes of form 6n + 1 oscillate largest numbers an infinite number of times. [Ribenboim] The proper factors of 6 are 1, 2 and 3. Note that 2*6^31, 2*6^3+1, 3*6^21 and 3*6^2+1 are all prime numbers. [Hartley] 6 + 1, 6 + 66 + 1, 6*66 + 1 , 6 + 66 + 666 + 1 and 6*66*666 + 1 are all primes. [Firoozbakht] The product of the first four nonzero Fibonacci numbers. Note that 6 + 1 and 6  1 are twin primes. This is the smallest such product of consecutive nonzero Fibonacci numbers. [Gupta] 6 is the smallest value for n such that n1, n+n^21, n+n^2+n^31, and n+n^2+n^3+n^41 are all primes. [Opao] ((6)*6*6  1, (6)*6*6 + 1 ) are twin primes. [Firoozbakht] All numbers between twin primes are evenly divisible by 6. [Smith] For n = 1, 2, 3, 4, 5^n + 6 is prime. [Rosenfeld] The lowest composite number containing distinct prime factors that cannot be expressed as the sum of two distinct prime numbers. [Capelle] The only integer which admits strictly fewer prime compositions (ordered partitions) than its predecessor. [Rupinski] The smallest invertible semiprime. [Capelle] The only semiprime whose divisors are consecutive numbers. [Silva] The smallest number whose factorial is a product of two factorials of primes (6! = 3!5!). Note that 6 is one of these factorials, in accordance with the equation (n!)! = n!(n!  1)! [Capelle] F_{0}*F_{1}*F_{2}*...*F_{6}*(F_{6}  1)  1 is prime, where F_{n} denotes the nth Fermat number. [Wesolowski] The number of prime Euler's "numeri idonei". [Capelle] 6 is probably the only even number n such that both numbers n^n  (n+1) and n^n + (n+1) are primes. [Firoozbakht] (1# + 2# + 3# + 4# + 5# + 6#) is onehalf of (1! + 2! + 3! + 4! + 5! + 6!). [Wesolowski] (6!) = 2! + 3! + 5!. [Wesolowski] The smallest number of primes that can be found on the Tetractys Puzzle. [Keith] The start of a sequence of invertible semiprimes (numbers when inverted give a different semiprime): 6, 9, 106, 119, 611, 901, ... . Can you find the first case where both reversals are also different semiprimes? Dare we call them "bemirpimes" (short for bidirectional emirpimes)? The last digit to appear in a Sophie Germain prime, first appearing in 641. The other 9 digits each first appear in smaller Sophie Germain primes. [Gaydos] There are 6 primes that can increase your conversion.
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