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m is a divisor of n (also called m divides n, or n is divisible by m) if all prime factors of m have at least the same multiplicity in n. The divisors of n are all products of some or all prime factors of n (including the empty product 1 of no prime factors). The number of divisors can be computed by increasing all multiplicities by 1 and then ...
The first few pairs of betrothed numbers (sequence A005276 in the OEIS) are: (48, 75), (140, 195), (1050, 1925), (1575, 1648), (2024, 2295), (5775, 6128). All known pairs of betrothed numbers have opposite parity. Any pair of the same parity must exceed 10 10.
All pairs of positive coprime numbers (m, n) (with m > n) can be arranged in two disjoint complete ternary trees, one tree starting from (2, 1) (for even–odd and odd–even pairs), [10] and the other tree starting from (3, 1) (for odd–odd pairs). [11] The children of each vertex (m, n) are generated as follows:
If none of its prime factors are repeated, it is called squarefree. (All prime numbers and 1 are squarefree.) For example, 72 = 2 3 × 3 2, all the prime factors are repeated, so 72 is a powerful number. 42 = 2 × 3 × 7, none of the prime factors are repeated, so 42 is squarefree. Euler diagram of numbers under 100:
However, amicable numbers where the two members have different smallest prime factors do exist: there are seven such pairs known. [8] Also, every known pair shares at least one common prime factor. It is not known whether a pair of coprime amicable numbers exists, though if any does, the product of the two must be greater than 10 65.
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More generally, a positive integer c is the hypotenuse of a primitive Pythagorean triple if and only if each prime factor of c is congruent to 1 modulo 4; that is, each prime factor has the form 4n + 1. In this case, the number of primitive Pythagorean triples (a, b, c) with a < b is 2 k−1, where k is the number of distinct prime factors of c ...