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One way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime (the factors need not be distinct, hence squares of primes are included). A composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to ...
Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry.
Fermat's little theorem states that if p is prime and a is coprime to p, then a p−1 − 1 is divisible by p. For an integer a > 1, if a composite integer x divides a x−1 − 1, then x is called a Fermat pseudoprime to base a. It follows that if x is a Fermat pseudoprime to base a, then x is coprime to a. Some sources use variations of this ...
When p is a prime, p 2 is a Fermat pseudoprime to base b if and only if p is a Wieferich prime to base b. For example, 1093 2 = 1194649 is a Fermat pseudoprime to base 2, and 11 2 = 121 is a Fermat pseudoprime to base 3. The number of the values of b for n are (For n prime, the number of the values of b must be n − 1, since all b satisfy the ...
In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli numbers .
A theorem states that n is prime if and only if all such functions p n are algebra endomorphisms. 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 ...
If the largest prime factor of a number is p then the number is B-smooth for any B ≥ p. In many scenarios B is prime, but composite numbers are permitted as well. A number is B-smooth if and only if it is p-smooth, where p is the largest prime less than or equal to B.
See also Theor.2.3 in ``Regularities of Twin, Triplet and Multiplet Prime Numbers," arXiv:1103.0447[math.NT], Global J.P.A.Math 8(2012), in press.) If the AP is prime for k consecutive values, then a must therefore be divisible by all primes p ≤ k. This also shows that an AP with common difference a cannot contain more consecutive prime terms ...