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Given an integer n, choose some integer a coprime to n and calculate a n − 1 modulo n. If the result is different from 1, then n is composite. If it is 1, then n may be prime. If a n −1 (modulo n) is 1 but n is not prime, then n is called a pseudoprime to base a. In practice, if a n −1 (modulo n) is 1, then n is usually prime.
The formulas show how to transform any right triangle with integer legs into another right triangle with integer legs whose hypotenuse is the square of the first triangle's hypotenuse. A Pythagorean prime is a prime number of the form 4 n + 1 {\displaystyle 4n+1} .
and // performs integer division or floor division, returning the floor of the quotient as an integer. In Python 2 (and most other programming languages), unless explicitly requested, x / y performed integer division, returning a float only if either input was a float.
More generally, given two Pythagorean triples (,,) and (,,) with largest entries c and e, one can join the corresponding triangles along the sides of length a (see the figure) for getting a Heronian triangle with side lengths ,, + and area (+) (this is an integer, since the area of a Pythagorean triangle is an integer).
Consequently, a square number is also triangular if and only if + is square, that is, there are numbers and such that =. This is an instance of the Pell equation x 2 − n y 2 = 1 {\displaystyle x^{2}-ny^{2}=1} with n = 8 {\displaystyle n=8} .
A former Allianz fund manager was spared prison time on Friday over his role in a meltdown of private investment funds sparked by the COVID-19 pandemic that caused an estimated $7 billion of ...
A 5-Con triangle pair is a pair of triangles that are similar but not congruent and that share three angles and two sidelengths. Primitive integer 5-Con triangles, in which the four distinct integer sides (two sides each appearing in both triangles, and one other side in each triangle) share no prime factor, have triples of sides
The two unidentified men were seen leaving the apartment at around 11:00 p.m., police said in the court documents, and Nichols was later seen leaving and coming back to the apartment several times ...