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The additive persistence of a number is smaller than or equal to the number itself, with equality only when the number is zero. For base b {\displaystyle b} and natural numbers k {\displaystyle k} and n > 9 {\displaystyle n>9} the numbers n {\displaystyle n} and n ⋅ b k {\displaystyle n\cdot b^{k}} have the same additive persistence.
An economical number has been defined as a frugal number, but also as a number that is either frugal or equidigital. gcd(m, n) (greatest common divisor of m and n) is the product of all prime factors which are both in m and n (with the smallest multiplicity for m and n).
The number of trailing zeros in a non-zero base-b integer n equals the exponent of the highest power of b that divides n. For example, 14000 has three trailing zeros and is therefore divisible by 1000 = 10 3, but not by 10 4. This property is useful when looking for small factors in integer factorization.
This means that 1 is a root of multiplicity 2, and −4 is a simple root (of multiplicity 1). The multiplicity of a root is the number of occurrences of this root in the complete factorization of the polynomial, by means of the fundamental theorem of algebra.
In number theory, the prime omega functions and () count the number of prime factors of a natural number . Thereby (little omega) counts each distinct prime factor, whereas the related function () (big omega) counts the total number of prime factors of , honoring their multiplicity (see arithmetic function).
The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. [3]
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An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio , ( 1 + 5 ) / 2 {\displaystyle (1+{\sqrt {5}})/2} , is an algebraic number, because it is a root of the polynomial x 2 − x − 1 .