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As an illustration of this, the parity cycle (1 1 0 0 1 1 0 0) and its sub-cycle (1 1 0 0) are associated to the same fraction 5 / 7 when reduced to lowest terms. In this context, assuming the validity of the Collatz conjecture implies that (1 0) and (0 1) are the only parity cycles generated by positive whole numbers (1 and 2 ...
Lothar Collatz (German:; July 6, 1910 – September 26, 1990) was a German mathematician, born in Arnsberg, Westphalia.. The "3x + 1" problem is also known as the Collatz conjecture, named after him and still unsolved.
The elements of a generating set of this semigroup are related to the sequence of numbers involved in the still open Collatz conjecture or the "3x + 1 problem". 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 3x + 1 semigroup was introduced by H. Farkas in ...
Conjecture Field Comments Eponym(s) Cites 1/3–2/3 conjecture: order theory: n/a: 70 abc conjecture: number theory: ⇔Granville–Langevin conjecture, Vojta's conjecture in dimension 1 ⇒Erdős–Woods conjecture, Fermat–Catalan conjecture Formulated by David Masser and Joseph Oesterlé. [1] Proof claimed in 2012 by Shinichi Mochizuki: n/a ...
The Erdős–Straus conjecture is one of many conjectures by Erdős, and one of many unsolved problems in mathematics concerning Diophantine equations. Although a solution is not known for all values of n , infinitely many values in certain infinite arithmetic progressions have simple formulas for their solution, and skipping these known values ...
The 1-factorization conjecture that if is odd or even and , respectively, then a -regular graph with vertices is 1-factorable. The perfect 1-factorization conjecture that every complete graph on an even number of vertices admits a perfect 1-factorization.
Bad idea, you should always include the loop cycle. True, it's easy to snip it for 3n+1 in the positive domain, but not so easy elsewhere. If you snip off the loop cycle of +(3n+1), there remains only one "trunk". But the -5 & -17 graphs in -(3n+1) have more than one "trunk", so removing the loop cycle makes the pieces of the remaining graph ...
In 2004, Ben J. Green and Terence Tao settled an old conjecture by proving the Green–Tao theorem: The primes contain arbitrarily long arithmetic progressions. [1] It follows immediately that there are infinitely many AP- k for any k .