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A prime p (where p ≠ 2, 5 when working in base 10) is called unique if there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 1/q. [8]
For instance, an even repeating cycle from an odd, prime reciprocal of ... The smallest prime number to yield such magic square in binary is 59 (111011 2), ...
3 is the second smallest prime number and the first odd prime number. It is the first unique prime, such that the period length value of 1 of the decimal expansion of its reciprocal, 0.333..., is unique. 3 is a twin prime with 5, and a cousin prime with 7, and the only known number such that ! − 1 and ! + 1 are prime, as well as the only ...
This is a list of articles about prime numbers. A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes.
a Markov number, appearing in solutions to the Markov Diophantine equation with other odd-indexed Fibonacci numbers. M 89 is the 10th Mersenne prime. Although 89 is not a Lychrel number in base 10, it is unusual that it takes 24 iterations of the reverse and add process to reach a palindrome. Among the known non-Lychrel numbers in the first ...
Twenty-three is also the next to last member of the first Cunningham chain of the first kind (2, 5, 11, 23, 47), [3] and the sum of the prime factors of the second set of consecutive discrete semiprimes, (21, 22). 23 is the smallest odd prime to be a highly cototient number, as the solution to () for the integers 95, 119, 143, and 529.
The sum of the reciprocals of all prime numbers diverges; that is: = + + + + + + + = This was proved by Leonhard Euler in 1737, [ 1 ] and strengthens Euclid 's 3rd-century-BC result that there are infinitely many prime numbers and Nicole Oresme 's 14th-century proof of the divergence of the sum of the reciprocals of the integers (harmonic series) .
special case of Fermat's little theorem, satisfied by all odd prime numbers ()solutions are called Wieferich primes (smallest example: 1093) ()satisfied by all prime numbers