DBpedia – Linked Data Fragments

Matches in DBpedia 2015-10 for { <http://dbpedia.org/resource/Proof_by_infinite_descent> ?p ?o }

• Proof_by_infinite_descent abstract "In mathematics, a proof by infinite descent is a particular kind of proof by contradiction which relies on the facts that the natural numbers are well ordered and that there are only a finite number of them that are smaller than any given one. One typical application is to show that a given equation has no solutions.Typically, one shows that if a solution to a problem existed, which in some sense was related to one or more natural numbers, it would necessarily imply that a second solution existed, which was related to one or more 'smaller' natural numbers. This in turn would imply a third solution related to smaller natural numbers, implying a fourth solution, therefore a fifth solution, and so on. However there cannot be an infinity of ever-smaller natural numbers, and therefore by mathematical induction (repeating the same step) the original premise—that any solution exists—must be incorrect. It is disproven because its logical outcome would require a contradiction.An alternative way to express this is to assume one or more solutions or examples exists. Then there must be a smallest solution or example—a minimal counterexample. We then prove that if a smallest solution exists, it must imply the existence of a smaller solution (in some sense)—which again proves that the existence of any solution would lead to a contradiction.The method of infinite descent was developed by Fermat, who often used it for Diophantine equations. Two typical examples are showing the non-solvability of the Diophantine equation r2 + s4 = t4 and proving Fermat's theorem on sums of two squares, which states that any prime p such that p ≡ 1 (mod 4) can be expressed as a sum of two squares (see proof). In some cases, to a modern eye, what he was using was (in effect) the doubling mapping on an elliptic curve. More precisely, his method of infinite descent was an exploitation in particular of the possibility of halving rational points on an elliptic curve E by inversion of the doubling formulae. The context is of a hypothetical rational point on E with large co-ordinates. Doubling a point on E roughly doubles the length of the numbers required to write it (as number of digits): so that a 'halved' point is quite clearly smaller. In this way Fermat was able to show the non-existence of solutions in many cases of Diophantine equations of classical interest (for example, the problem of four perfect squares in arithmetic progression).".
• Proof_by_infinite_descent wikiPageExternalLink sq_root.shtml.
• Proof_by_infinite_descent wikiPageID "358069".
• Proof_by_infinite_descent wikiPageLength "13527".
• Proof_by_infinite_descent wikiPageOutDegree "50".
• Proof_by_infinite_descent wikiPageRevisionID "682400558".
• Proof_by_infinite_descent wikiPageWikiLink Abelian_variety.
• Proof_by_infinite_descent wikiPageWikiLink Ad_infinitum.
• Proof_by_infinite_descent wikiPageWikiLink Algebra.
• Proof_by_infinite_descent wikiPageWikiLink Algebraic_number_theory.
• Proof_by_infinite_descent wikiPageWikiLink Ancient_Greek.
• Proof_by_infinite_descent wikiPageWikiLink André_Weil.
• Proof_by_infinite_descent wikiPageWikiLink Arithmetic_progression.
• Proof_by_infinite_descent wikiPageWikiLink Category:Diophantine_equations.
• Proof_by_infinite_descent wikiPageWikiLink Category:Mathematical_proofs.
• Proof_by_infinite_descent wikiPageWikiLink Category:Mathematical_terminology.
• Proof_by_infinite_descent wikiPageWikiLink Contradiction.
• Proof_by_infinite_descent wikiPageWikiLink Coprime.
• Proof_by_infinite_descent wikiPageWikiLink Coprime_integers.
• Proof_by_infinite_descent wikiPageWikiLink Diophantine_equation.
• Proof_by_infinite_descent wikiPageWikiLink Elliptic_curve.
• Proof_by_infinite_descent wikiPageWikiLink Fermat.
• Proof_by_infinite_descent wikiPageWikiLink Fermats_Last_Theorem.
• Proof_by_infinite_descent wikiPageWikiLink Fermats_theorem_on_sums_of_two_squares.
• Proof_by_infinite_descent wikiPageWikiLink Finitely-generated_abelian_group.
• Proof_by_infinite_descent wikiPageWikiLink Finitely_generated_abelian_group.
• Proof_by_infinite_descent wikiPageWikiLink Galois_cohomology.
• Proof_by_infinite_descent wikiPageWikiLink Glossary_of_arithmetic_and_Diophantine_geometry.
• Proof_by_infinite_descent wikiPageWikiLink Hippasus.
• Proof_by_infinite_descent wikiPageWikiLink Integer_factorization.
• Proof_by_infinite_descent wikiPageWikiLink Inversive_geometry.
• Proof_by_infinite_descent wikiPageWikiLink Irrational_number.
• Proof_by_infinite_descent wikiPageWikiLink John_Horton_Conway.
• Proof_by_infinite_descent wikiPageWikiLink L-function.
• Proof_by_infinite_descent wikiPageWikiLink Louis_J._Mordell.
• Proof_by_infinite_descent wikiPageWikiLink Mathematical_induction.
• Proof_by_infinite_descent wikiPageWikiLink Mathematics.
• Proof_by_infinite_descent wikiPageWikiLink Minimal_counterexample.
• Proof_by_infinite_descent wikiPageWikiLink Modular_arithmetic.
• Proof_by_infinite_descent wikiPageWikiLink Mordell.
• Proof_by_infinite_descent wikiPageWikiLink Mordell–Weil_theorem.
• Proof_by_infinite_descent wikiPageWikiLink Natural_number.
• Proof_by_infinite_descent wikiPageWikiLink Natural_numbers.
• Proof_by_infinite_descent wikiPageWikiLink Number_theory.
• Proof_by_infinite_descent wikiPageWikiLink Pierre_de_Fermat.
• Proof_by_infinite_descent wikiPageWikiLink Prime_factorization.
• Proof_by_infinite_descent wikiPageWikiLink Proof_by_contradiction.
• Proof_by_infinite_descent wikiPageWikiLink Proofs_of_Fermats_theorem_on_sums_of_two_squares.
• Proof_by_infinite_descent wikiPageWikiLink Pythagorean_triangle.
• Proof_by_infinite_descent wikiPageWikiLink Pythagorean_triple.
• Proof_by_infinite_descent wikiPageWikiLink Pythagoreanism.
• Proof_by_infinite_descent wikiPageWikiLink Rational_number.
• Proof_by_infinite_descent wikiPageWikiLink Rational_point.
• Proof_by_infinite_descent wikiPageWikiLink Real_number.
• Proof_by_infinite_descent wikiPageWikiLink Square_number.
• Proof_by_infinite_descent wikiPageWikiLink Square_root_of_2.
• Proof_by_infinite_descent wikiPageWikiLink Vieta_jumping.
• Proof_by_infinite_descent wikiPageWikiLink Well-order.
• Proof_by_infinite_descent wikiPageWikiLink Well-ordering_principle.
• Proof_by_infinite_descent wikiPageWikiLinkText "Proof by infinite descent".
• Proof_by_infinite_descent wikiPageWikiLinkText "descent for elliptic curves".
• Proof_by_infinite_descent wikiPageWikiLinkText "infinite descent".
• Proof_by_infinite_descent wikiPageWikiLinkText "method of descent".
• Proof_by_infinite_descent wikiPageWikiLinkText "method of infinite descent".
• Proof_by_infinite_descent wikiPageWikiLinkText "one proof here".
• Proof_by_infinite_descent wikiPageWikiLinkText "proof by infinite descent".
• Proof_by_infinite_descent hasPhotoCollection Proof_by_infinite_descent.
• Proof_by_infinite_descent title "Example of Fermat's last theorem".
• Proof_by_infinite_descent title "Infinite descent".
• Proof_by_infinite_descent urlname "ExampleOfFermatsLastTheorem".
• Proof_by_infinite_descent urlname "InfiniteDescent".
• Proof_by_infinite_descent wikiPageUsesTemplate Template:Harvtxt.
• Proof_by_infinite_descent wikiPageUsesTemplate Template:Math.
• Proof_by_infinite_descent wikiPageUsesTemplate Template:PlanetMath.
• Proof_by_infinite_descent wikiPageUsesTemplate Template:Reflist.
• Proof_by_infinite_descent wikiPageUsesTemplate Template:See_also.
• Proof_by_infinite_descent subject Category:Diophantine_equations.
• Proof_by_infinite_descent subject Category:Mathematical_proofs.
• Proof_by_infinite_descent subject Category:Mathematical_terminology.
• Proof_by_infinite_descent hypernym Kind.
• Proof_by_infinite_descent type Proof.
• Proof_by_infinite_descent type Theorem.
• Proof_by_infinite_descent type Thing.
• Proof_by_infinite_descent comment "In mathematics, a proof by infinite descent is a particular kind of proof by contradiction which relies on the facts that the natural numbers are well ordered and that there are only a finite number of them that are smaller than any given one.".
• Proof_by_infinite_descent label "Proof by infinite descent".
• Proof_by_infinite_descent seeAlso Fermats_right_triangle_theorem.
• Proof_by_infinite_descent sameAs نزول_غير_منته.
• Proof_by_infinite_descent sameAs Mètode_del_descens_infinit.
• Proof_by_infinite_descent sameAs Unendlicher_Abstieg.
• Proof_by_infinite_descent sameAs Descenso_infinito.
• Proof_by_infinite_descent sameAs Méthode_de_descente_infinie.
• Proof_by_infinite_descent sameAs נסיגה_אינסופית.
• Proof_by_infinite_descent sameAs Végtelen_leszállás.
• Proof_by_infinite_descent sameAs Discesa_infinita.
• Proof_by_infinite_descent sameAs 無限降下法.
• Proof_by_infinite_descent sameAs 무한강하법.
• Proof_by_infinite_descent sameAs Bewijs_door_oneindige_afdaling.
• Proof_by_infinite_descent sameAs m.01_3hv.
• Proof_by_infinite_descent sameAs Метод_бесконечного_спуска.
• Proof_by_infinite_descent sameAs Dokaz_z_neskončnim_spustom.

• next