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More specifically, the Millennium Prize version of the conjecture is that, if the elliptic curve E has rank r, then the L-function L(E, s) associated with it vanishes to order r at s = 1. Hilbert's tenth problem dealt with a more general type of equation, and in that case it was proven that there is no algorithmic way to decide whether a given ...
For any integer n, n ≡ 1 (mod 2) if and only if 3n + 1 ≡ 4 (mod 6). Equivalently, n − 1 / 3 ≡ 1 (mod 2) if and only if n ≡ 4 (mod 6). Conjecturally, this inverse relation forms a tree except for the 1–2–4 loop (the inverse of the 4–2–1 loop of the unaltered function f defined in the Statement of the problem section of ...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Such questions are usually more difficult to solve than regular mathematical exercises like "5 − 3", even if one knows the mathematics required to solve the problem. Known as word problems , they are used in mathematics education to teach students to connect real-world situations to the abstract language of mathematics.
The 3x + 1 semigroup has been used to prove a weaker form of the Collatz conjecture. In fact, it was in such context the concept of the 3 x + 1 semigroup was introduced by H. Farkas in 2005. [ 2 ] Various generalizations of the 3 x + 1 semigroup have been constructed and their properties have been investigated.
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The term "algebraic equation" dates from the time when the main problem of algebra was to solve univariate polynomial equations. This problem was completely solved during the 19th century; see Fundamental theorem of algebra, Abel–Ruffini theorem and Galois theory. Since then, the scope of algebra has been dramatically enlarged.
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