In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that...

Goldbach dated 12 April 1749. The proof relies on infinite descent, and is only briefly sketched in the letter. The full proof consists in five steps and is...

contradiction. Law of excluded middle Law of noncontradiction Proof by exhaustion Proof by infinite descent Modus tollens Reductio ad absurdum Bishop, Errett 1967...

{\sqrt {2}}} . One proof of the number's irrationality is the following proof by infinite descent. It is also a proof of a negation by refutation: it proves...

traditionally called proof by infinite descent. In which case, there may be multiple and more complex ways to structure the argument of the proof. The assumption...

Barbara, and Dolan. For one proof by infinite descent, see Infinite descent#Non-solvability of r2 + s4 = t4. Fermat's proof demonstrates that no right...

including proof by infinite descent. Direct proof methods include proof by exhaustion and proof by induction. A direct proof is the simplest form of proof there...

ample use of a related principle: indirect proof by infinite descent. The induction hypothesis was also employed by the Swiss Jakob Bernoulli, and from then...

covers number theory, including a proof that the square root of two is irrational and a proof that there are infinitely many prime numbers. Further advances...

but his proof by infinite descent contained a major gap. However, since Euler himself had proved the lemma necessary to complete the proof in other work...

is an irrational number. Below are two short proofs of irrationality: This is a proof by infinite descent. Recall that: the whole is the longer part plus...

431–442. doi:10.1137/0204037. Retrieved 2022-12-11. More generally, proof by infinite descent is applicable to any well-ordered set. Hardy and Wright, p. 42...

factorization method—Fermat's factorization method—and popularized the proof by infinite descent, which he used to prove Fermat's right triangle theorem which...

Fermat's marginal notes with the proof of the right triangle theorem, in 1670. Fermat's proof is a proof by infinite descent. It shows that, from any example...

variations of Vieta jumping, all of which involve the common theme of infinite descent by finding new solutions to an equation using Vieta's formulas. Vieta...

{\displaystyle g(x)=1} and f η ( x ) {\displaystyle f_{\eta }(x)} with infinitely spikes towards infinity but still integrable. One such example could be...

defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. Cantor's method of proof of this...

approximation of gradient descent optimization, since it replaces the actual gradient (calculated from the entire data set) by an estimate thereof (calculated...

been known as far back as the seventeenth century. The process of infinite descent of Fermat was well known, but Mordell succeeded in establishing the...

best known for his role as Ethan Gross on the ABC drama series Body of Proof, Matt Mahoney on the CBS political drama series Madam Secretary, and a young...

arithmetic, after the examples provided by Gödel's incompleteness theorem and Gerhard Gentzen's 1943 direct proof of the unprovability of ε0-induction in...

βράχιστος χρόνος (brákhistos khrónos) 'shortest time'), or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where...

determined by the priorities appearing in the play. Typically, player 0 wins an infinite play if the largest priority that occurs infinitely often in the...

gradient descent in the infinite-width limit is fully equivalent to kernel gradient descent with the NTK. As a result, using gradient descent to minimize...

is characterized by absurdity, meaninglessness, and despair. According to this argument, humans are finite beings living in an infinite universe, and their...

in the case when the interval [ a , b ] {\displaystyle [a,b]} is infinite). The proof proceeds otherwise as above, but with a slightly different approximation...

April 12 – Euler produces the first proof of Fermat's theorem on sums of two squares, based on infinite descent. April 12 – Official opening of the Radcliffe...

w_{i}\in \mathbb {R} ^{d}} to a possibly infinite dimensional feature represented by a kernel K {\displaystyle K} by instead performing the recursion on the...

aware of this). He stated that a proof could be found by infinite descent. Fermat stated and proved (by infinite descent) in the appendix to Observations...

proof that there are infinitely many pairs of prime numbers that differ by less than 70 million. This result implies the existence of an infinitely repeatable...