In number theory, a Carmichael number is a composite number , which in modular arithmetic satisfies the congruence relation:
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann.
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, 10 = 2 ⋅ 5 is square-free, but 18 = 2 ⋅ 3 ⋅ 3 is not, because 18 is divisible by 9 = 32. The smallest positive square-free numbers are
2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultures.
A highly composite number is a positive integer with more divisors than any smaller positive integer has. A related concept is that of a largely composite number, a positive integer which has at least as many divisors as any smaller positive integer. The name can be somewhat misleading, as the first two highly composite numbers are not actually composite numbers; however, all further terms are.
In mathematics, a multiply perfect number is a generalization of a perfect number.
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. It is denoted by π(x) (unrelated to the number π).
43 (forty-three) is the natural number following 42 and preceding 44.
109 is the natural number following 108 and preceding 110.
In number theory, the Mertens function is defined for all positive integers n as
1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.
In analytic number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens.
In number theory, a practical number or panarithmic number is a positive integer such that all smaller positive integers can be represented as sums of distinct divisors of . For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2.
A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are
In number theory, a colossally abundant number is a natural number that, in a particular, rigorous sense, has many divisors. Particularly, it is defined by a ratio between the sum of an integer's divisors and that integer raised to a power higher than one. For any such exponent, whichever integer has the highest ratio is a colossally abundant number. It is a stronger restriction than that of a superabundant number, but not strictly stronger than that of an abundant number.
In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes converges to a finite value known as Brun's constant, usually denoted by B2. Brun's theorem was proved by Viggo Brun in 1919, and it has historical importance in the introduction of sieve methods.
In mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev functionϑ (x) or θ (x) is given by
277 is the natural number following 276 and preceding 278.
In mathematics, the sophomore's dream is the pair of identities
