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 * 56 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*6^3-1, 2*6^3+1, 3*6^2-1 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 n-1, n+n^2-1, n+n^2+n^3-1, and n+n^2+n^3+n^4-1 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]
F0*F1*F2*...*F6*(F6
- 1) - 1 is prime, where Fn 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 one-half of (1! + 2! + 3! + 4! + 5! + 6!). [Wesolowski]
(6!) = 2! + 3! + 5!. [Wesolowski]
The first perfect number.
There are only six prime-digit primes with distinct odd digits, i.e., 3, 5, 7, 37, 53, 73. [Loungrides]
The smallest number of primes that can be found on the
Tetractys Puzzle. [Keith]
6 is the product of the first two primes. [Jeursen]
The only perfect oblong number, i.e., 6=2*3. [Loungrides]
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 bi-directional 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.
In mathematics, 6 is considered to be the first ‘perfect’ number. [Oliveira]
(There are 16 curios for this number that have not yet been approved by an editor.)
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