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It follows that in computer algebra, many algorithms involving integer exponents must be changed when the exponentiation bases do not commute. Some general purpose computer algebra systems use a different notation (sometimes ^^ instead of ^ ) for exponentiation with non-commuting bases, which is then called non-commutative exponentiation .
Bernoulli's inequality can be proved for case 2, in which is a non-negative integer and , using mathematical induction in the following form: we prove the inequality for r ∈ { 0 , 1 } {\displaystyle r\in \{0,1\}} ,
The property of whether an integer is even (or not) is known as its parity. If two numbers are both even or both odd, they have the same parity. By contrast, if one is even and the other odd, they have different parity. The addition, subtraction and multiplication of even and odd integers obey simple rules.
holds for every integer a coprime to n. In algebraic terms, λ(n) is the exponent of the multiplicative group of integers modulo n. As this is a finite abelian group, there must exist an element whose order equals the exponent, λ(n). Such an element is called a primitive λ-root modulo n.
However, due to the multivalued nature of complex power functions for non-integer exponents, one must be careful to specify the branch of the complex logarithm being used. In addition, no matter which branch is used, if c {\displaystyle c} is not a positive integer, then the function is not differentiable at 0.
The exponent of the term is =, and this sum can be interpreted as a representation of as a partition into copies of each number . Therefore, the number of terms of the product that have exponent n {\displaystyle n} is exactly p ( n ) {\displaystyle p(n)} , the same as the coefficient of x n {\displaystyle x^{n}} in the sum on the left.
Values of t such that 2 t is an integer are all of the form t = log 2 m for some integer m, while for 3 t to be an integer, t must be of the form t = log 3 n for some integer n. By setting x 1 = 1, x 2 = t , y 1 = log(2), and y 2 = log(3), the four exponentials conjecture implies that if t is irrational then one of the following four numbers is ...
In elementary number theory, the lifting-the-exponent lemma (LTE lemma) provides several formulas for computing the p-adic valuation of special forms of integers. The lemma is named as such because it describes the steps necessary to "lift" the exponent of p {\displaystyle p} in such expressions.