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In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. [1] [2] [3] Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to ...
Although not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a "proof without words". The left-hand picture below is an example of a historic visual proof of the Pythagorean theorem in the case of the (3,4,5) triangle.
The conjecture (Conjecture 5.1 in [29]) states that if the ideal boundary of a word-hyperbolic group G is homeomorphic to the 2-sphere, then G admits a properly discontinuous cocompact isometric action on the hyperbolic 3-space (so that G is essentially a 3-dimensional Kleinian group).
A conjecture is a proposition that is unproven. Conjectures are related to hypotheses , which in science are empirically testable conjectures. In mathematics , a conjecture is an unproven proposition that appears correct.
In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, [1] is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be. It states that
As reformulated, it became the "paving conjecture" for Euclidean spaces, and then a question on random polynomials, in which latter form it was solved affirmatively. 2015: Jean Bourgain, Ciprian Demeter, and Larry Guth: Main conjecture in Vinogradov's mean-value theorem: analytic number theory: Bourgain–Demeter–Guth theorem, ⇐ decoupling ...
Goldbach's weak conjecture, every odd number greater than 5 can be expressed as the sum of three primes, is a consequence of Goldbach's conjecture. Ivan Vinogradov proved it for large enough n (Vinogradov's theorem) in 1937, [1] and Harald Helfgott extended this to a full proof of Goldbach's weak conjecture in 2013. [2] [3] [4]
Dejean's theorem (formerly Dejean's conjecture) is a statement about repetitions in infinite strings of symbols. It belongs to the field of combinatorics on words ; it was conjectured in 1972 by Françoise Dejean and proven in 2009 by Currie and Rampersad and, independently, by Rao.