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In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. [1] For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are ...
Then E. T. Parker found a counterexample of order 10 using a one-hour computer search. Finally Parker, Bose, and Shrikhande showed this conjecture to be false for all n ≥ 10. In 1798 A. M. Legendre claimed that 6 is not the sum of 2 rational cubes, [9] which as Lamé pointed out in 1865 is false as 6 = (37/21) 3 + (17/21) 3.
Conjectures disproven through counterexample are sometimes referred to as false conjectures (cf. the Pólya conjecture and Euler's sum of powers conjecture). In the case of the latter, the first counterexample found for the n=4 case involved numbers in the millions, although it has been subsequently found that the minimal counterexample is ...
For example, if there were an even integer N = p + 1 larger than 4, for p a prime, that could not be expressed as the sum of two primes in the modern sense, then it would be a counterexample to the modern version of the third conjecture (without being a counterexample to the original version).
One example is the parallel postulate, which is neither provable nor refutable from the remaining axioms of Euclidean geometry. Mathematicians have shown there are many statements that are neither provable nor disprovable in Zermelo–Fraenkel set theory with the axiom of choice (ZFC), the standard system of set theory in mathematics (assuming ...
Although the discovery of counterexamples by Boyce and Huneke meant that the decade-long pursuit of a proof of the commuting function conjecture was lost, it did enable researchers to focus their efforts on investigating under what conditions, in addition to the ones already discovered, the conjecture still might hold true. [2]
For example, for 2 5 a + 1 there are 3 increases as 1 iterates to 2, 1, 2, 1, ... For the special purpose of searching for a counterexample to the Collatz conjecture, ...
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 ...