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The Pólya conjecture states that for any n > 1, if the natural numbers less than or equal to n (excluding 0) are partitioned into those with an odd number of prime factors and those with an even number of prime factors, then the former set has at least as many members as the latter set. Repeated prime factors are counted repeatedly; for ...
Negative numbers: Real numbers that are less than zero. Because zero itself has no sign, neither the positive numbers nor the negative numbers include zero. When zero is a possibility, the following terms are often used: Non-negative numbers: Real numbers that are greater than or equal to zero. Thus a non-negative number is either zero or positive.
It follows from the definition that each natural number is equal to the set of all natural numbers less than it. This definition, can be extended to the von Neumann definition of ordinals for defining all ordinal numbers, including the infinite ones: "each ordinal is the well-ordered set of all smaller ordinals."
[3] [4] Later that year, James Maynard announced a related breakthrough which proved that there are infinitely many prime gaps of some size less than or equal to 600. [5] As of April 14, 2014, one year after Zhang's announcement, according to the Polymath project wiki , n has been reduced to 246. [ 6 ]
A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. ... Number of odd entries in row ... A prime number that is ...
A subset A of positive integers has natural density α if the proportion of elements of A among all natural numbers from 1 to n converges to α as n tends to infinity.. More explicitly, if one defines for any natural number n the counting function a(n) as the number of elements of A less than or equal to n, then the natural density of A being α exactly means that [1]
A perfect totient number is an integer that is equal to the sum of its iterated totients. That is, we apply the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and add together the resulting sequence of numbers; if the sum equals n, then n is a perfect totient number.
Demonstration, with Cuisenaire rods, of the first four highly composite numbers: 1, 2, 4, 6. A highly composite number is a positive integer that has more divisors than all smaller positive integers. If d(n) denotes the number of divisors of a positive integer n, then a positive integer N is highly composite if d(N) > d(n) for all n < N.