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PhET Interactive Simulations is part of the University of Colorado Boulder which is a member of the Association of American Universities. [10] The team changes over time and has about 16 members consisting of professors, post-doctoral students, researchers, education specialists, software engineers (sometimes contractors), educators, and administrative assistants. [11]
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm that, for any given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns), can decide whether the equation has a solution with all unknowns taking integer values.
This theorem forms the basis for Wiener's attack, a polynomial-time exploit of the RSA cryptographic protocol that can occur for an injudicious choice of public and private keys (specifically, this attack succeeds if the prime factors of the public key n = pq satisfy p < q < 2p and the private key d is less than (1/3)n 1/4). [7]
So equivalence is defined by an integer Möbius transformation on the real numbers, or by a member of the Modular group (), the set of invertible 2 × 2 matrices over the integers. Each rational number is equivalent to 0; thus the rational numbers are an equivalence class for this relation.
Factorization is the process of splitting a mathematical object, often integers or polynomials, into a product of factors. Fermat's last theorem Fermat's last theorem, one of the most famous and difficult to prove theorems in number theory, states that for any integer n > 2, the equation a n + b n = c n has no positive integer solutions.
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers , finite fields , and function fields .
This is a linear Diophantine equation, related to Bézout's identity. + = + The smallest nontrivial solution in positive integers is 12 3 + 1 3 = 9 3 + 10 3 = 1729.It was famously given as an evident property of 1729, a taxicab number (also named Hardy–Ramanujan number) by Ramanujan to Hardy while meeting in 1917. [1]
The name algebraic integer comes from the fact that the only rational numbers that are algebraic integers are the integers, and because the algebraic integers in any number field are in many ways analogous to the integers. If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O K.