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Amicable numbers. Amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, s (a)= b and s (b)= a, where s (n)=σ (n)- n is equal to the sum of positive divisors of n except n itself (see also divisor function). The smallest pair of amicable ...
For example, the "primitive" friendly pair 6 and 28 gives rise to friendly pairs 6n and 28n for all n that are congruent to 1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, or 41 modulo 42. [4] This shows that the natural density of the friendly numbers (if it exists) is positive.
Euler's factorization method. Euler 's factorization method is a technique for factoring a number by writing it as a sum of two squares in two different ways. For example the number can be written as or as and Euler's method gives the factorization . The idea that two distinct representations of an odd positive integer may lead to a ...
In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B. [1] In terms of set-builder notation, that is [2][3] A table can be created by taking the Cartesian product of a set of rows and a set of columns.
Ruth–Aaron pair. In mathematics, a Ruth–Aaron pair consists of two consecutive integers (e.g., 714 and 715) for which the sums of the prime factors of each integer are equal: 714 = 2 × 3 × 7 × 17, 715 = 5 × 11 × 13, and. 2 + 3 + 7 + 17 = 5 + 11 + 13 = 29. There are different variations in the definition, depending on how many times to ...
The third largest prime factor is greater than 100, [28] and less than . [29] N has at least 101 prime factors and at least 10 distinct prime factors. [21] [30] If 3 is not one of the factors of N, then N has at least 12 distinct prime factors. [31] N is of the form
A set of integers can also be called coprime if its elements share no common positive factor except 1. A stronger condition on a set of integers is pairwise coprime, which means that a and b are coprime for every pair (a, b) of different integers in the set. The set {2, 3, 4} is coprime, but it is not pairwise coprime since 2 and 4 are not ...
The multiplicity of a prime factor p of n is the largest exponent m for which p m ... A Ruth-Aaron pair is two consecutive numbers (x ... 18, 20, 22, 24, 26, 28, 30 ...