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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 following is a list of all 52 currently known (as of January 2025) Mersenne primes and corresponding perfect numbers, along with their exponents p. The largest 18 of these have been discovered by the distributed computing project Great Internet Mersenne Prime Search , or GIMPS; their discoverers are listed as "GIMPS / name ", where the name ...
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.
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.
To find the number of negative roots, change the signs of the coefficients of the terms with odd exponents, i.e., apply Descartes' rule of signs to the polynomial = + + This polynomial has two sign changes, as the sequence of signs is (−, +, +, −) , meaning that this second polynomial has two or zero positive roots; thus the original ...
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.
In mathematics, Kummer's theorem is a formula for the exponent of the highest power of a prime number p that divides a given binomial coefficient. In other words, it gives the p-adic valuation of a binomial coefficient. The theorem is named after Ernst Kummer, who proved it in a paper, (Kummer 1852).