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Mathematical proof technique using contradiction
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
Proof_by_infinite_descent
Argument that leads to a logical absurdity
original assumption is false, and √2 is irrational. Proof by infinite descent is a method of proof whereby a smallest object with desired property is shown
Reductio_ad_absurdum
Condition under which an odd prime is a sum of two squares
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
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Unique positive real number which when multiplied by itself gives 2
{\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
Square_root_of_2
Way of arriving to a mathematical proof
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
Direct_proof
Partial results found before the complete proof
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
Proof of Fermat's Last Theorem for specific exponents
Proof_of_Fermat's_Last_Theorem_for_specific_exponents
Smallest example which falsifies a claim
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
Minimal_counterexample
Reasoning for mathematical statements
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
Mathematical_proof
Form of mathematical proof
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
Mathematical_induction
Category of mathematical proof
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
Proof_of_impossibility
Generalization of "n-th" to infinite cases
is true for natural numbers is the basis of Fermat's method of proof by infinite descent, which can be generalized to ordinals and other well-ordered classes
Ordinal_number
17th-century conjecture proved by Andrew Wiles in 1994
Fermat's proof. Only one relevant proof by Fermat has survived, in which he uses the technique of infinite descent to show that the area of a right triangle
Fermat's_Last_Theorem
French mathematician and lawyer (1601–1665)
method — Fermat's factorization method — and popularized the proof by infinite descent, which he used to prove Fermat's right triangle theorem which
Pierre_de_Fermat
Number, approximately 1.618
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
Golden_ratio
Rational right triangles cannot have square area
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
Fermat's right triangle theorem
Fermat's_right_triangle_theorem
Theorem of probability theory
{N}}(0,I)} . This form has applications in Stein variational gradient descent and Stein variational policy gradient. The univariate probability density
Stein's_lemma
Mathematical proof technique
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
Vieta_jumping
The group of K-rational points of an abelian variety is a finitely-generated abelian group
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
Mordell–Weil_theorem
leads to D n > D n + 1 {\displaystyle D_{n}>D_{n+1}} , setting up an infinite descent of positive integers ( D n > D n + 1 > D n + 2 > ⋯ > 0 {\displaystyle
Proof_that_pi_is_irrational
Optimization algorithm
approximation of gradient descent optimization, since it replaces the actual gradient (calculated from the entire data set) by an estimate thereof (calculated
Stochastic_gradient_descent
American actor (born 1978)
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
Geoffrey_Arend
Mathematician (1845–1918)
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
Georg_Cantor
Type of kernel induced by artificial neural networks
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
Neural_tangent_kernel
Mathematical game played on a directed graph
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
Parity_game
Fastest curve descent without friction
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
Brachistochrone_curve
Philosophical question
is characterized by absurdity, meaninglessness, and despair. According to this argument, humans are finite beings living in an infinite universe, and their
Existence_of_God
1638 book by Galileo Galilei
) He resolves the contradiction by denying the possibility of comparing infinite numbers (and of comparing infinite and finite numbers): We can only
Two_New_Sciences
Method of machine learning
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
Online_machine_learning
Integers have unique prime factorizations
divided evenly by some prime number.) Proposition 31 is proved directly by infinite descent. Any number either is prime or is measured by some prime number
Fundamental theorem of arithmetic
Fundamental_theorem_of_arithmetic
Method for approximate evaluation of integrals
{\displaystyle a} and b {\displaystyle b} may be infinite. This technique was originally presented in the book by Laplace (1774). In Bayesian statistics, Laplace's
Laplace's_method
Every natural number can be represented as the sum of four integer squares
which rp is also the sum of four squares (this is in the spirit of the infinite descent method of Fermat). For this purpose, we consider for each xi the yi
Lagrange's four-square theorem
Lagrange's_four-square_theorem
loop in a computer program by well-founded descent. A basic property of a well-founded relation is that it has no infinite descending chains. Therefore
Loop_variant
American bassist (born 1968)
success of 1995's Mellon Collie and the Infinite Sadness, Corgan said that she began an "apparent slow descent into insanity and/or drugs (take your pick)
D'arcy_Wretzky
German philosopher (1788–1860)
example in the starry heavens, suggests to imagination the infinite, which in turn leads by subtle turns of contemplation to the concept of God. Schopenhauer's
Arthur_Schopenhauer
Type of infinite structure
specifically in model theory, an infinite structure ( M , < , … ) {\displaystyle (M,<,\dots )} that is totally ordered by < {\displaystyle <} is called an
O-minimal_theory
Argument for the existence of God
itself. Other variations expound its finitude in the impossibility of an infinite regress of causal interactions, kinetic interactions between objects in
Cosmological_argument
Class of reinforcement learning algorithms
starting state, and T {\displaystyle T} is the time-horizon (which can be infinite). The policy gradient is defined as ∇ θ J ( θ ) {\displaystyle \nabla _{\theta
Policy_gradient_method
Statement that all non empty subsets of positive numbers contains a least element
criminal" method and is similar in its nature to Fermat's method of "infinite descent". Garrett Birkhoff and Saunders Mac Lane wrote in A Survey of Modern
Well-ordering_principle
How many times a number is divisible by 2
classic proof that the square root of 2 is irrational operates by infinite descent. Usually, the descent part of the proof is abstracted away by assuming
Singly_and_doubly_even
American actor and marine (born 1971)
multiplayer. Reyes wrote the book, Hero Living: Seven Strides to Awaken Your Infinite Power (2009), with Angela Smith. He has also authored articles for OFFGRID
Rudy_Reyes_(actor)
Polynomial equation whose integer solutions are sought
for proving that there is no solution— is the method of infinite descent, which was introduced by Pierre de Fermat. Another general method is the Hasse
Diophantine_equation
Conjectures connecting number theory and geometry
generalized these to automorphic cuspidal representations, which are certain infinite dimensional irreducible representations of the general linear group GL(n)
Langlands_program
Film genre depicting survival efforts
Variety. Retrieved October 11, 2018. Ehrlich, David (March 23, 2022). "'Infinite Storm' Review: Naomi Watts Tries to Save an Unwilling Hiker in Tedious
Survival_film
Ancient Indian sage
towards the Moon God. His descendants have a direct descent from the Chandravamsha and a matrilineal descent from the Suryavamsha. Śhāṇḍilya is the son of the
Shandilya_(Rishi)
British mathematician (1922–2015)
of the foundations of the modern theory of infinite descent. His best-known single result may be the proof that the Tate-Shafarevich group of an elliptic
J._W._S._Cassels
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
1749_in_science
British mathematician and broadcaster (born 1984)
(Vsauce), that started in November 2025. The same month, she presented The Infinite Explorer, a six-part National Geographic documentary series. The following
Hannah_Fry
there infinitely many balanced primes? Are there infinitely many cluster primes? Are there infinitely many cousin primes? Are there infinitely many Cullen
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Compact astronomical body
should have a central singularity, where the curvature of spacetime is infinite. Objects whose gravitational fields are too strong for light to escape
Black_hole
Mathematical optimization method
rates is infinite and the sum of squares of learning rates is finite) of diminishing learning rate scheme (see section "Stochastic gradient descent") and
Backtracking_line_search
logic. Infinite descent Infinite descent was Pierre de Fermat's classical method for Diophantine equations. It became one half of the standard proof of the
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Mathematician and philosopher (1906–1978)
machine Gödel Prize Gödel t-norm Gödel's ontological proof Gödel's Dialectica interpretation Infinite-valued logic List of Austrian scientists List of pioneers
Kurt_Gödel
Hungarian mathematician (1913–1996)
resulted in the marriage of George and Esther Szekeres. By the time he was 20, Erdős had found a proof for Bertrand's postulate. In 1934, at the age of 21
Paul_Erdős
Construct in mathematics
conjecture: When a second descent exists, the number of generators found is an even number less than what is indicated by the first descent. Cassels explores
Selmer_group
Open problem on 3x+1 and x/2 functions
shown to hold for all positive integers up to 2.36×1021, but no general proof has been found. It is named after the mathematician Lothar Collatz, who
Collatz_conjecture
Ten emanations in Kabbalah
of emanations from the infinite divine essence. This was necessary due to the inability of humanity to exist in God's infinite presence. God does not
Sefirot
Group comohology of Galois modules
already implicit in the infinite descent arguments of Fermat for elliptic curves. Numerous direct calculations were done, and the proof of the Mordell–Weil
Galois_cohomology
American actress and singer
consecutive wins for the Screen Actors Guild Award for Outstanding Performance by an Ensemble in a Comedy Series. Guerrero is the author of In the Country We
Diane_Guerrero
Hungarian and American mathematician and physicist (1903–1957)
were proven during the creation and proof of the above theorems, particularly regarding distributivity (such as infinite distributivity), von Neumann developing
John_von_Neumann
Iterative method for finding maximum likelihood estimates in statistical models
published by C. F. Jeff Wu in 1983. Wu's proof established the EM method's convergence also outside of the exponential family, as claimed by Dempster–Laird–Rubin
Expectation–maximization algorithm
Expectation–maximization_algorithm
American musician (born 1967)
He is of English, Irish, and Scottish descent on his father's side, and Belgian, Flemish, and Italian descent on his mother's side. He grew up Catholic
Billy_Corgan
Fictional DC Comics character
published by DC Comics, commonly in association with Batman. The substantial character is portrayed as a powerful mob boss and gangster of Italian descent in
Sal_Maroni
Comics character
The Titans Academy is later attacked by Deathstroke and his forces during the events of Dark Crisis on Infinite Earths. Raven and the other Titans fight
Raven_(DC_Comics)
Canadian rapper (born 1980)
진실을 요구합니다2), which has a membership of over 33,000 netizens despite proof provided by both the university and the police that Tablo did indeed graduate
Tablo
Statistical algorithm
stochastic gradient descent method in that the filter is only adapted based on the error at the current time. It was invented in 1960 by Stanford University
Least_mean_squares_filter
Chinese-born American mathematician
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
Yitang_Zhang
Mathematical optimization algorithm
Townsend presents a Lean verified proof of the conjecture for all restart lengths. Consider the linear system Ax = b given by A x = [ 4 1 1 3 ] [ x 1 x 2 ]
Conjugate_gradient_method
British-Lebanese mathematician (1929–2019)
did not have a proof of the periodicity theorem, a proof along similar lines was shortly afterwards found by R. Wood. He found a proof of several generalizations
Michael_Atiyah
{\displaystyle Y} and Z {\displaystyle Z} , and utilizes stochastic gradient descent and other optimization algorithms for training. The fig illustrates the
Deep backward stochastic differential equation method
Deep_backward_stochastic_differential_equation_method
{\displaystyle \alpha t=\pi } , the frequency of spikes is zero since the period is infinite since θ {\displaystyle \theta } can no longer pass through θ = π {\displaystyle
Theta_model
automatically updated combination of the twenty lists of Latin phrases starting by a given letter. The classical Latin alphabet did not have the letter "J".
List_of_Latin_phrases_(full)
Season of television series
season of the One Piece anime television series is produced by Toei Animation and directed by Tatsuya Nagamine, Satoshi Itō and Yasunori Koyama. The season
One_Piece_season_20
or suppressed to hide the fact that it is instead a disc, or a single infinite plane. The conspiracy often implicates NASA. Other claims include that
List_of_conspiracy_theories
Number line and triangular tiling's symmetry mathematical structure
only finitely many possibilities for the number of descents of an affine permutation, but infinitely many affine permutations, it is not possible to naively
Affine_symmetric_group
Filipino actress and singer (born 1971)
presented on Finding Your Roots revealed that she is of Filipino and German descent, with maternal ancestry traced to the Prussian region. Her great great
Lea_Salonga
Conjecture in number theory
Theorem. Since 1997, Beal has offered a monetary prize for a peer-reviewed proof of this conjecture or a counterexample. The value of the prize has increased
Beal_conjecture
Howell, Maddy (May 14, 2026). "Seahaven unleash introspective single "Infinite Blue"". Rock Sound. Retrieved May 14, 2026. "Napalm Death bassist Shane
List_of_2026_albums
(March 21, 2008). "IDW Publishing to Release Everybody's Dead, a Zombie Comedy by the Writer of Best-Selling Angel: After the Fall". Comics Bulletin. Retrieved
List of IDW Publishing publications
List_of_IDW_Publishing_publications
Algebraic curve in mathematics
smaller than any constant exist on E. The proof of the theorem is thus a variant of the method of infinite descent and relies on the repeated application
Elliptic_curve
British novelist and literary critic (born 1954)
December 2023. Crown, Sarah (16 August 2023). "Caret by Adam Mars-Jones review – a semi-infinite novel". The Guardian. ISSN 0261-3077. Retrieved 10 September
Adam_Mars-Jones
Australian and American mathematician (born 1975)
conditions on when there exist infinitely many n × 1 matrices x such that all components of Ax + v are prime numbers.[GT10] The proof of Green and Tao was incomplete
Terence_Tao
Subpermutation of a longer permutation
from either of these by interchanging the last two elements or the 1 and the 2. Because this collection of permutations is infinite (in fact, it is the
Permutation_pattern
American media personality (born 1981)
Paris (2020), Cooking with Paris (2021), Paris in Love (2021–2023), and Infinite Icon: A Visual Memoir (2026). Under her company, 11:11 Media, she has launched
Paris_Hilton
Philosophical tradition in Muslim culture
impossibility of completing an actual infinite by successive addition", states: "An actual infinite cannot be completed by successive addition." "The temporal
Islamic_philosophy
Class of artificial neural network
perceptron network is equivalent to an infinitely deep feedforward network. Similar networks were published by Kaoru Nakano in 1971, Shun'ichi Amari in
Recurrent_neural_network
Racist foundations of Nazism
Aryan certificate, which was typically done by providing proof that their four grandparents were of Aryan descent and obtaining an Ahnenpass (ancestor’s pass)
Nazi_racial_theories
Group of comic book supervillains
grown to mural size and is now installed in Dayton Manor in Prague. In Infinite Crisis, a member of H.I.V.E. mentioned seeing Punch and Jewelee at a "Save
Brotherhood_of_Dada
Ladybug and was founded by Master Fu. Marinette Dupain-Cheng is a 14-year-old student of French, Chinese, and Italian descent at Françoise Dupont High
List of Miraculous: Tales of Ladybug & Cat Noir characters
List_of_Miraculous:_Tales_of_Ladybug_&_Cat_Noir_characters
minor planet discoveries are confirmed, they are given a permanent number by the IAU's Minor Planet Center (MPC), and the discoverers can then submit names
Meanings of minor-planet names: 10001–11000
Meanings_of_minor-planet_names:_10001–11000
Group in which the order of every element is a power of p
smaller example H/Z whose normalizer in G/Z is N/Z = H/Z, creating an infinite descent. As a corollary, every finite p-group is nilpotent. In another direction
P-group
Formula for refraction angles
of refraction. Brachistochrone curve – Fastest curve descent without friction for a simple proof by Jacob Bernoulli Calculus of variations – Differential
Snell's_law
Directed graph with no directed cycles
one vertex to another vertex. A walk in a directed graph is a (finite or infinite) sequence ( v 1 , v 2 , … ) {\displaystyle (v_{1},v_{2},\dotsc )} of vertices
Directed_acyclic_graph
Form of divine energy in Hindu mysticism
"willed process" of raising the Kundalini; rather, the centers are opened by a "descent of the Divine Consciousness" from above. He explained that while certain
Kundalini
American mathematician
first study of surjunctive groups and a short proof of the De Bruijn–Erdős theorem on coloring infinite graphs. As well as being a research mathematician
Walter_Gottschalk
French mathematician (1906-1998)
theorem had an ad hoc proof; Weil began the separation of the infinite descent argument into two types of structural approach, by means of height functions
André_Weil
Mechanism of evolution by differential reproduction
considering the infinite complexity of the relations of all organic beings to each other and to their conditions of existence, causing an infinite diversity
Natural_selection
Triangle whose side lengths and area are integers
Robert Daniel (1915). "I. Introduction. Rational Triangles. Method of Infinite Descent". Diophantine Analysis. Wiley. pp. 1–23. Dickson, Leonard Eugene (1920)
Heronian_triangle
Mathematical statistics distance measure
other hand, on the logit scale implied by weight of evidence, the difference between the two is enormous – infinite perhaps; this might reflect the difference
Kullback–Leibler_divergence
Conspiracy theory about first family of the United States
political controversies regarding aid given to Haiti by the Clinton Foundation, such as "hurricane-proof" classroom trailers that were found to be structurally
Clinton body count conspiracy theory
Clinton_body_count_conspiracy_theory
Elementary is an American crime drama created by Robert Doherty and loosely based on Sherlock Holmes and other characters appearing in the works of Sir
List_of_Elementary_episodes
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PROOF BY-INFINITE-DESCENT
PROOF BY-INFINITE-DESCENT
PROOF BY-INFINITE-DESCENT
PROOF BY-INFINITE-DESCENT
PROOF BY-INFINITE-DESCENT
PROOF BY-INFINITE-DESCENT
PROOF BY-INFINITE-DESCENT
PROOF BY-INFINITE-DESCENT
PROOF BY-INFINITE-DESCENT
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