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The first: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600 (sequence A000142 in the OEIS). 0! = 1 is sometimes included. A k-smooth number (for a natural number k) has its prime factors ≤ k (so it is also j-smooth for any j > k). m is smoother than n if the largest prime factor of m is below the largest of n.
The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b).. In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
There are known formulae to evaluate the prime-counting function (the number of primes smaller than a given value) faster than computing the primes. This has been used to compute that there are 1,925,320,391,606,803,968,923 primes (roughly 2 × 10 21 ) smaller than 10 23 .
As of December 2019 the largest number known to have been factored by a general-purpose algorithm is RSA-240, which has 240 decimal digits (795 bits) and is the product of two large primes. [154] Shor's algorithm can factor any integer in a polynomial number of steps on a quantum computer. [155]
RSA-150 has 150 decimal digits (496 bits), and was withdrawn from the challenge by RSA Security. RSA-150 was eventually factored into two 75-digit primes by Aoki et al. in 2004 using the general number field sieve (GNFS), years after bigger RSA numbers that were still part of the challenge had been solved. The value and factorization are as ...
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Some of the small groups that do not have a normal p-complement include: Order 24: The symmetric group S 4; Order 48: The binary octahedral group and the product S 4 × Z 2; Order 60: The alternating group A 5. The smallest order for which it is not known how many nonisomorphic groups there are is 2048 = 2 11. [7]
Vice President Kamala Harris won California and its 54 electoral votes by 20 percentage points, with The Associated Press calling it for her as polls closed at 8 p.m. Pacific time on Election Day.