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The natural logarithms of all positive integers lower than 23 are known to have binary BBP-type formulae. [19] 23 is the first prime p for which unique factorization of cyclotomic integers based on the pth root of unity breaks down. [20] 23 is the smallest positive solution to Sunzi's original formulation of the Chinese remainder theorem.
All integers are either even or odd. A square has even multiplicity for all prime factors (it is of the form a 2 for some a). The first: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 (sequence A000290 in the OEIS). A cube has all multiplicities divisible by 3 (it is of the form a 3 for some a).
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.
The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are even, and decimal numbers that end in 0 or 5 are divisible by 5. [11] The set of all primes is sometimes denoted by (a boldface capital P) [12] or by (a blackboard bold capital P). [13]
The tables below list all of the divisors of the numbers 1 to 1000. A divisor of an integer n is an integer m, for which n/m is again an integer (which is necessarily also a divisor of n). For example, 3 is a divisor of 21, since 21/7 = 3 (and therefore 7 is also a divisor of 21). If m is a divisor of n, then so is −m. The tables below only ...
The following is a list of all currently known Mersenne primes and perfect numbers, along with their corresponding exponents p. As of 2024, there are 52 known Mersenne primes (and therefore perfect numbers), the largest 18 of which have been discovered by the distributed computing project Great Internet Mersenne Prime Search, or GIMPS. [2]
If one of the factors is composite, it can in turn be written as a product of smaller factors, for example 60 = 3 · 20 = 3 · (5 · 4). Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem.
1, 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, ... For n ≥ 2, a ( n ) is the prime that is finally reached when you start with n , concatenate its prime factors (A037276) and repeat until a prime is reached; a ( n ) = −1 if no prime is ever reached.