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The multiplicity of a prime factor p of n is the largest exponent m for which ... A Ruth-Aaron pair is two consecutive numbers ... 84: 2 2 ·3·7 85: 5·17 86: 2·43 ...
Two numbers with the same "abundancy" form a friendly pair; ... The sum of an integer's unique factors, up to n=2000. ... 84: 84/83 84: 224: 8/3 85: 108: 108/85 86: ...
The same prime factor may occur more than once; this example has two copies of the prime factor When a prime occurs multiple times, exponentiation can be used to group together multiple copies of the same prime number: for example, in the second way of writing the product above, 5 2 {\displaystyle 5^{2}} denotes the square or second power of 5 ...
However, any two least common multiples of the same pair of elements are associates. [10] In a unique factorization domain , any two elements have a least common multiple. [ 11 ] In a principal ideal domain , the least common multiple of a and b can be characterised as a generator of the intersection of the ideals generated by a and b [ 10 ...
Any Ruth–Aaron pair of square-free integers belongs to both lists with the same sum of prime factors. The intersection also contains pairs that are not square-free, for example (7129199, 7129200) = (7×11 2 ×19×443, 2 4 ×3×5 2 ×13×457).
If p is an irregular prime and p divides the numerator of the Bernoulli number B 2k for 0 < 2k < p − 1, then (p, 2k) is called an irregular pair. In other words, an irregular pair is a bookkeeping device to record, for an irregular prime p , the particular indices of the Bernoulli numbers at which regularity fails.
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The notion of greatest common divisor can more generally be defined for elements of an arbitrary commutative ring, although in general there need not exist one for every pair of elements. [ 26 ] If R is a commutative ring, and a and b are in R , then an element d of R is called a common divisor of a and b if it divides both a and b (that is, if ...