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The tables contain the prime factorization of the natural numbers from 1 to 1000. When n is a prime number, the prime factorization is just n itself, written in bold below. The number 1 is called a unit. It has no prime factors and is neither prime nor composite.
The first 1000 prime numbers ... write the prime factorization of n in base 10 and concatenate the factors; iterate until a prime is reached. 2, 3, 211, 5, 23, 7 ...
The difficulty of computing φ(n) without knowing the factorization of n is thus the difficulty of computing d: this is known as the RSA problem which can be solved by factoring n. The owner of the private key knows the factorization, since an RSA private key is constructed by choosing n as the product of two (randomly chosen) large primes p and q.
The size of the input to the algorithm is log 2 n or the number of bits in the binary representation of n. Any element of the order n c for a constant c is exponential in log n . The running time of the number field sieve is super-polynomial but sub-exponential in the size of the input.
A definite bound on the prime factors is possible. Suppose P i is the i 'th prime, so that P 1 = 2, P 2 = 3, P 3 = 5, etc. Then the last prime number worth testing as a possible factor of n is P i where P 2 i + 1 > n; equality here would mean that P i + 1 is a factor. Thus, testing with 2, 3, and 5 suffices up to n = 48 not just 25 because the ...
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. To factorize a small integer n using mental or pen-and-paper arithmetic, the simplest method is trial division : checking if the number is divisible by prime numbers 2 ...
A prime sieve or prime number sieve is a fast type of algorithm for finding primes. There are many prime sieves. The simple sieve of Eratosthenes (250s BCE), the sieve of Sundaram (1934), the still faster but more complicated sieve of Atkin [1] (2003), sieve of Pritchard (1979), and various wheel sieves [2] are most common.
In number theory, a frugal number is a natural number in a given number base that has more digits than the number of digits in its prime factorization in the given number base (including exponents). [1] For example, in base 10, 125 = 5 3, 128 = 2 7, 243 = 3 5, and 256 = 2 8 are frugal numbers (sequence A046759 in the OEIS).