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From this we see that r is any even integer and that s and t are factors of r 2 /2. All Pythagorean triples may be found by this method. When s and t are coprime, the triple will be primitive. A simple proof of Dickson's method has been presented by Josef Rukavicka, J. (2013). [7] Example: Choose r = 6. Then r 2 /2 = 18. The three factor-pairs ...
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 ...
The members of S are called generators of F S, and the number of generators is the rank of the free group. An arbitrary group G is called free if it is isomorphic to F S for some subset S of G , that is, if there is a subset S of G such that every element of G can be written in exactly one way as a product of finitely many elements of S and ...
A general-purpose factoring algorithm, also known as a Category 2, Second Category, or Kraitchik family algorithm, [10] has a running time which depends solely on the size of the integer to be factored. This is the type of algorithm used to factor RSA numbers. Most general-purpose factoring algorithms are based on the congruence of squares method.
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 relatively prime.
Algorithm: SFF (Square-Free Factorization) Input: A monic polynomial f in F q [x] where q = p m Output: Square-free factorization of f R ← 1 # Make w be the product (without multiplicity) of all factors of f that have # multiplicity not divisible by p c ← gcd(f, f′) w ← f/c # Step 1: Identify all factors in w i ← 1 while w ≠ 1 do y ...
How to calculate a factor rate. Using the factor rate provided by the lender, you can quickly calculate the cost of the borrowed funds. For example, if you borrowed $100,000 with a factor rate of ...
The example at left is that of an orthogonal array with symbol set {1,2} and strength 2. Notice that the four ordered pairs (2-tuples) formed by the rows restricted to the first and third columns, namely (1,1), (2,1), (1,2) and (2,2), are all the possible ordered pairs of the two element set and each appears exactly once.