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The first 15 superior highly composite numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 (sequence A002201 in the OEIS) are also the first 15 colossally abundant numbers, which meet a similar condition based on the sum-of-divisors function rather than the number of divisors. Neither ...
Every composite number can be written as the product of two or more (not necessarily distinct) primes. [2] For example, the composite number 299 can be written as 13 × 23, and the composite number 360 can be written as 2 3 × 3 2 × 5; furthermore, this representation is unique up to the order of the factors.
the k given prime numbers p i must be precisely the first k prime numbers (2, 3, 5, ...); if not, we could replace one of the given primes by a smaller prime, and thus obtain a smaller number than n with the same number of divisors (for instance 10 = 2 × 5 may be replaced with 6 = 2 × 3; both have four divisors);
In-between these two conditions lies the definition of Carmichael number of order m for any positive integer m as any composite number n such that p n is an endomorphism on every Z n-algebra that can be generated as Z n-module by m elements. Carmichael numbers of order 1 are just the ordinary Carmichael numbers.
A gauge number denotes a thickness on a standardized scale which, for most purposes, starts at 20g (0.812 mm thick— often used for the posts for nose studs), and increases in thickness (as the gauge number decreases) to 0g, then 00g, and rarely goes any further as these thicknesses come closer and closer to 1 ⁄ 2 ″.
For example, if a = 5 + 3i, and b = 2 – 8i, one has N(a) = 34, N(b) = 68, and N(a + b) = 74. As the greatest common divisor of the three norms is 2, the greatest common divisor of a and b has 1 or 2 as a norm. As a gaussian integer of norm 2 is necessary associated to 1 + i, and as 1 + i divides a and b, then the greatest common divisor is 1 + i.
If the Euler's criterion formula is used modulo a composite number, the result may or may not be the value of the Jacobi symbol, and in fact may not even be −1 or 1. For example, For example, ( 19 45 ) = 1 and 19 45 − 1 2 ≡ 1 ( mod 45 ) .
There are only 21853 pseudoprimes base 2 that are less than 2.5 × 10 10 (see page 1005 of [3]). This means that, for n up to 2.5 × 10 10, if 2 n −1 (modulo n) equals 1, then n is prime, unless n is one of these 21853 pseudoprimes. Some composite numbers (Carmichael numbers) have the property that a n − 1 is 1 (modulo n) for every a that ...