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Conjecture in number theory
The abc conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and
Abc_conjecture
17th-century conjecture proved by Andrew Wiles in 1994
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that there are no positive integers a
Fermat's_Last_Theorem
Theorem about consecutive perfect powers
x^{\lambda n}} uniformly in m and n. The general conjecture would follow from the ABC conjecture. Pillai's conjecture means that for every natural number n, there
Catalan's_conjecture
Seven mathematical problems with a US$1 million prize for each solution
unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem
Millennium_Prize_Problems
Conjecture in number theory
Szpiro's conjecture relates to the conductor and the discriminant of an elliptic curve. In a slightly modified form, it is equivalent to the well-known abc conjecture
Szpiro's_conjecture
Mathematical theory by Shinichi Mochizuki
to provide a proof for various outstanding conjectures in number theory, in particular the abc conjecture. Mochizuki and a few other mathematicians claim
Inter-universal Teichmüller theory
Inter-universal_Teichmüller_theory
Aharoni-Korman conjecture also known as the fishbone conjecture Atiyah conjecture (not a conjecture to start with) Borsuk's conjecture Bunkbed conjecture Chinese
List_of_conjectures
Japanese mathematician
from non-mathematicians due to claims it provides a resolution of the abc conjecture. Shinichi Mochizuki was born to parents Kiichi and Anne Mochizuki. When
Shinichi_Mochizuki
Conjecture in number theory
The Beal conjecture is the following conjecture in number theory: Unsolved problem in mathematics If A x + B y = C z {\displaystyle A^{x}+B^{y}=C^{z}}
Beal_conjecture
Generalization of the abc conjecture to more than three integers
number theory, the n conjecture is a conjecture stated by Browkin & Brzeziński (1994) as a generalization of the abc conjecture to more than three integers
N_conjecture
Product of the prime factors of an integer
prime}}}p} The radical plays a central role in the statement of the abc conjecture. Radical numbers for the first few positive integers are 1, 2, 3, 2
Radical_of_an_integer
On heights of points on algebraic varieties over number fields
Vojta's conjecture is a conjecture introduced by Paul Vojta (1987) about heights of points on algebraic varieties over number fields. The conjecture was motivated
Vojta's_conjecture
Does the plane contains a dense set of points whose distances are all rational
counterexample to the Bombieri–Lang conjecture and to the abc conjecture. It would also solve Harborth's conjecture, on the existence of drawings of planar
Erdős–Ulam_problem
Topics referred to by the same term
ABC, abc, or Abc in Wiktionary, the free dictionary. ABC, abc, or Abc commonly refers to: ABC, the first three letters of the Latin script ABC, abc,
ABC
some Zariski-closed set. n conjecture: a generalization of the abc conjecture to more than three integers. abc conjecture: for any ε > 0 {\displaystyle
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
There are at most a finite number of consecutive powers
number of solutions. The truth of Pillai's conjecture, in turn, would follow from the truth of the abc conjecture. Narkiewicz, Wladyslaw (2011), Rational
Tijdeman's_theorem
In mathematics, when is n!+1 a square
follow from the abc conjecture that there are only finitely many Brown numbers. More generally, it would also follow from the abc conjecture that n ! + A
Brocard's_problem
Unsolved problem in mathematics
weak form of Hall's conjecture would follow from the ABC conjecture. A generalization to other perfect powers is Pillai's conjecture, though it is also
Hall's_conjecture
French mathematician (born 1954)
a French mathematician who, along with David Masser, formulated the abc conjecture which has been called "the most important unsolved problem in diophantine
Joseph_Oesterlé
Generalization of Fermat's Last Theorem and of Catalan's conjecture,
theory, the Fermat–Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture. The conjecture states that the equation
Fermat–Catalan_conjecture
Result concerning properties of Galois representations associated with modular forms
Ribet's theorem (earlier called the epsilon conjecture or ε-conjecture) is part of number theory. It concerns properties of Galois representations associated
Ribet's_theorem
Polish mathematician
together with Juliusz Brzeziński, he formulated the n-conjecture—a version of the abc conjecture involving n > 2 integers. "Zmarł Profesor Jerzy Browkin
Jerzy_Browkin
Prime such that p^2 divides 2^(p-1)-1
numbers as well as more general subjects such as number fields and the abc conjecture. As of 2026[update], the only known Wieferich primes are 1093 and 3511
Wieferich_prime
Canadian mathematician
numerous contributions to number theory, in particular to work on the abc conjecture. In 1976 he obtained, with Alan Baker, an effective improvement to Liouville's
Cameron_Leigh_Stewart
Theorem about polynomials, analogous to the abc conjecture for integers
theorem, is a mathematical theorem about polynomials, analogous to the abc conjecture for integers. It is named after Walter Wilson Stothers, who published
Mason–Stothers_theorem
Study of algorithms for performing number theoretic computations
investigate conjectures and open problems in number theory, including the Riemann hypothesis, the Birch and Swinnerton-Dyer conjecture, the ABC conjecture, the
Computational_number_theory
Top 0–9 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z abc conjecture The abc conjecture of Masser and Oesterlé attempts to state as much as possible
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Potential counterexample to the generalized Riemann hypothesis
Granville and Stark showed that a certain uniform formulation of the abc conjecture for number fields implies "no Siegel zeros" for negative discriminants
Siegel_zero
Non-profit BOINC based volunteer computing project
ABC@Home was an educational and non-profit network computing project finding abc-triples related to the abc conjecture in number theory using the Berkeley
ABC@Home
Top 0–9 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z abc conjecture The abc conjecture says that for all ε > 0, there are only finitely many coprime
Glossary_of_number_theory
Numbers whose prime factors all divide the number more than once
its smallest term must be congruent to 7, 27, or 35 modulo 36. If the abc conjecture is true, there are only a finite number of sets of three consecutive
Powerful_number
Diophantine geometry. With Joseph Oesterlé in 1985, Masser formulated the abc conjecture, which has been called "the most important unsolved problem in Diophantine
David_Masser
Elliptic curves
Gauss–Manin connection may give some important hints for Vojta's conjecture, ABC conjecture and so on; in 2012, he published his Inter-universal Teichmuller
Hodge–Arakelov_theory
[K:\mathbb {Q} ]} .) It is known that the abc conjecture implies Lang's conjecture, and that the analogue of Lang's conjecture over one dimensional characteristic
Néron–Tate_height
Japanese idol girl group
Albums". Billboard Japan. 29 November 2023. Retrieved 28 April 2026. ABC Conjecture: "Hot Albums". Billboard Japan. 17 December 2025. Archived from the
22/7_(group)
Japanese mathematician
Riemann Hypothesis, Langlands conjecture, Sato conjecture, 2017. (Japanese) with Shinya Koyama, Introduction to the ABC conjecture, 2018. (Japanese) Ishikura
Nobushige_Kurokawa
Theoretical object in mathematics
These approximations imply solutions to important problems like the abc conjecture. The extensions of F1 later on were denoted as Fq with q = 1n. Together
Field_with_one_element
American mathematician
diophantine analysis in the circle of ideas around the Mordell conjecture and abc conjecture. This suggested the importance of the integer solutions (affine
Paul_Vojta
Product of numbers from 1 to n
{\displaystyle 16!=14!\cdot 5!\cdot 2!} . It would follow from the abc conjecture that there are only finitely many nontrivial examples. The greatest
Factorial
French mathematician (1941–2020)
a conjecture (now known as Szpiro's conjecture) relating the discriminant of an elliptic curve with its conductor. His conjecture inspired the abc conjecture
Lucien_Szpiro
Number without repeated prime factors
{x}}} by x + c x 1 / 5 log x . {\displaystyle x+cx^{1/5}\log x.} The abc conjecture would allow x + x o ( 1 ) {\displaystyle x+x^{o(1)}} . The squarefree
Square-free_integer
American mathematician (born 1947)
also made contributions to the understanding of Siegel zeroes, to the ABC conjecture, to modular forms on GL ( n ) {\displaystyle \operatorname {GL} (n)}
Dorian_M._Goldfeld
Fermat's Last Theorem Mordell conjecture Euler's sum of powers conjecture abc Conjecture Catalan's conjecture Pillai's conjecture Hasse principle Diophantine
List_of_number_theory_topics
Mathematical arithmetic dynamics function
\alpha \in K} . In the other direction, Juul et al. proved that if the abc conjecture holds for number fields, K {\displaystyle K} is a number field and f
Arboreal Galois representation
Arboreal_Galois_representation
Open source middleware system for volunteer and grid computing
Retrieved 2022-10-01. Cruncher Pete (2011-09-02). "Information on Collatz Conjecture". Archived from the original on 2013-12-26. Retrieved 2012-02-03. "BOINC
Berkeley Open Infrastructure for Network Computing
Berkeley_Open_Infrastructure_for_Network_Computing
On prime divisors of differences two nth powers
Math. 539 (2001), 75-122 J.H. Silverman, Wieferich's criterion and the abc-conjecture, J. Number Theory 30 (1988), 226-237 P. Ingram, J.H. Silverman, Uniform
Zsigmondy's_theorem
Relates rational elliptic curves to modular forms
statement was known as the Taniyama–Shimura conjecture, Taniyama–Shimura–Weil conjecture, or the modularity conjecture for elliptic curves. The theorem states
Modularity_theorem
Mathematical functions that quantify complexity
and Q {\displaystyle \mathbb {Q} } ). abc conjecture Birch and Swinnerton-Dyer conjecture Elliptic Lehmer conjecture Heath-Brown–Moroz constant Height of
Height_function
British mathematician
Mason-Stothers theorem) in 1981. This is an analogue of the well-known abc conjecture for integers: indeed it was the inspiration for the latter. Later independent
Walter_Wilson_Stothers
"Information on Collatz Conjecture". Retrieved 2012-02-03. "Collatz Conjecture". 2012. Retrieved 2012-01-13. "BOINCstats — Collatz Conjecture". boincstats.com
List of volunteer computing projects
List_of_volunteer_computing_projects
released online a series of papers in which he claims to prove the abc conjecture. Despite the publication in a peer-reviewed journal later, his proof
List_of_incomplete_proofs
German-American artificial intelligence researcher
He co-founded EleutherAI and founded the AI safety research company Conjecture, which he led as CEO until 2026. He is US Director of ControlAI, a nonprofit
Connor_Leahy
Russian mathematician (1937–2010)
the theory of symplectic topology as a distinct discipline. The Arnold conjecture on the number of fixed points of Hamiltonian symplectomorphisms and Lagrangian
Vladimir_Arnold
Class of integer sequences in mathematics
(electronic), 2001. J. H. Silverman. Wieferich's criterion and the abc-conjecture. J. Number Theory, 30(2):226–237, 1988. B. Poonen. Using elliptic curves
Elliptic divisibility sequence
Elliptic_divisibility_sequence
Brazilian mathematician
with André Neves, he proved the Willmore conjecture. Since then, among proving other important conjectures, Marques and Neves greatly extended Almgren–Pitts
Fernando_Codá_Marques
Elliptic curve associated with a Fermat triple
conjecture or ε-conjecture. In the summer of 1986, Ribet (1990) proved the epsilon conjecture, thereby proving that the Taniyama–Shimura–Weil conjecture implies
Frey_curve
Proto-Indo-European water fire deity
emerging from the water, golden, and "clothed in lightning", which has been conjectured to be a reference to fire. His regular identification with Agni, who
*H₂epom_Nepōts
Video-sharing platform
to extremist videos, little systematic evidence exists to support this conjecture", and that such exposure was "heavily concentrated among a small group
YouTube
Mathematics award
was found in 1993. In 2006, Grigori Perelman, who proved the Poincaré conjecture, refused his Fields Medal, stated "I'm not interested in money or fame;
Fields_Medal
Theorem about triangles
theorem about triangles in the Euclidean plane. In its original form, as conjectured by Tôda Ono (小野藤太) in 1914, the inequality is actually false; however
Ono's_inequality
Dense arrangement of congruent spheres in an infinite, regular arrangement
conjecture states that this is the highest density that can be achieved by any arrangement of spheres, either regular or irregular. This conjecture was
Close-packing of equal spheres
Close-packing_of_equal_spheres
JFK assassination conspiracy theory
involves American conspirators. The secretive nature of the CIA, and the conjecture surrounding the high-profile political assassinations in the United States
CIA Kennedy assassination conspiracy theory
CIA_Kennedy_assassination_conspiracy_theory
Theory with particles of spin more than two
topic is due to the AdS/CFT correspondence where there is a number of conjectures relating higher-spin theories to weakly coupled conformal field theories
Higher-spin_theory
American politician (1934–2022)
George W. Bush. After the defeat of Utah's Senator Bob Bennett in 2010, conjecture began as to whether six-term Senator Hatch would retire in 2012. It was
Orrin_Hatch
Country in Eastern Europe and North Asia
Clay Millennium Prize Problems Award for his final proof of the Poincaré conjecture in 2002, as well as the Fields Medal in 2006. Alexander Popov was among
Russia
Quantum field theory
Problems". Here the prize-problem consists, especially, in a proof of the conjecture that the lowest excitations of a pure Yang–Mills theory (i.e. without
Yang–Mills_theory
fish, mass in reptiles is frequently poorly documented, thus subject to conjecture and estimation. The saltwater crocodile is considered to be the largest
List_of_largest_reptiles
Compact astronomical body
hair conjecture proposes that dynamic gravitational collapse always results in an object characterized with only these three properties. The conjecture is
Black_hole
French mathematician and politician (born 1973)
Carlo Cercignani. Villani, in fact, proved the so-called Cercignani's conjecture. His second lecture at the Royal Institution is titled 'The Extraordinary
Cédric_Villani
General relativity model near spacetime singularities
for appropriately defined spatial scale factors. This is called the BKL conjecture. For most types of matter the effect of the matter fields on the dynamics
BKL_singularity
1956 scientific conference on artificial intelligence
in Hanover, New Hampshire. The study is to proceed on the basis of the conjecture that every aspect of learning or any other feature of intelligence can
Dartmouth_workshop
American murderer (1936–2001)
2010. "Henry Lee Lucas Dies in Prison". abcnews.go.com. Huntsville, Texas: ABC News. March 13, 2001. Retrieved July 12, 2010. Ramsland, Katherine. "Henry
Henry_Lee_Lucas
professor of mathematics, solves two long-standing problems, the Height Zero Conjecture and the Deligne-Lusztig theory. Mathematicians believe that it may lead
2024_in_science
Phenomena of sensory perception
such examples as having one's hair cut or combed. This has led to the conjecture that ASMR might be related to the act of grooming. For example, David
ASMR
Queen of England from 1536 to 1537
Catherine Parr, convinced him to do so. One non-contemporary source conjectures that she may have been pregnant and had a miscarriage by Christmas 1536
Jane_Seymour
Ancient Roman amphitheater in Rome
historical grounds for the supposition, other than the reasonably plausible conjecture that some of the many martyrs may well have been Christian. Later popes
Colosseum
English theoretical physicist (1942–2018)
1992, Hawking conjectured that travel into the past is effectively impossible. On 28 June 2009, as a tongue-in-cheek test of the conjecture, he held a party
Stephen_Hawking
List of U.S.-backed regime change operations
supplied to them. Precisely how these lists had been compiled is a matter of conjecture, but it is certain that some of the Ba'th leaders were in touch with American
United States involvement in regime change
United_States_involvement_in_regime_change
Species of tropical plant
species. However, as no other wild specimens have been found, this is still conjecture. The plant is grown widely throughout Southeast Asia and South Asia. The
Pandanus_amaryllifolius
Ruth Behar writes that Mexican Inquisition cases "hint at a fascinating conjecture of sexuality, witchcraft, and religion, in which Spanish, indigenous,
Witchcraft_in_Latin_America
Decade of the Gregorian calendar (2000–2009)
he proved Thurston's geometrization conjecture. This consequently solved in the affirmative the Poincaré conjecture, posed in 1904, which before its solution
2000s
large reptiles is frequently poorly documented and many are subject to conjecture and estimation. Dinosaurs are now extinct, except for birds, which are
Largest_and_heaviest_animals
American actress (born 1954)
role of Annie "Rebel" Bello in the ABC drama series Rebel, which was written by Krista Vernoff. Due to low ratings, ABC cancelled Rebel after airing five
Katey_Sagal
American political figure (born 1969)
such [intentional] misconduct. The Court cannot accept speculation or conjecture in place of clear and convincing evidence". The judge further ruled that
Kari_Lake
Largest species of toothed whale
sperm whale's eyes afford good vision and sensitivity to light. They conjectured that sperm whales use vision to hunt squid, either by detecting silhouettes
Sperm_whale
1813 novel by Jane Austen
original manuscript, study of the first drafts of the novel is reduced to conjecture. From the large number of letters in the final novel, it is assumed that
Pride_and_Prejudice
Practices believed to use supernatural powers
Ruth Behar writes that Mexican Inquisition cases "hint at a fascinating conjecture of sexuality, witchcraft, and religion, in which Spanish, indigenous,
Witchcraft
Leader of the Soviet Union from 1922 to 1924
Truth and Method (1960) The Structure of Scientific Revolutions (1962) Conjectures and Refutations (1963) One-Dimensional Man (1964) Knowledge and Human
Vladimir_Lenin
give him 51 starts. Some sources attribute the record at 148 due to some conjecture over whether De Cesaris had 147 or 148 retirements; De Cesaris ran out
List of Formula One driver records
List_of_Formula_One_driver_records
Skyscraper in Manhattan, New York
Kaszycki & Sons Contractors, Inc., 874 F.2d 912 (2nd Cir. 1989) ("Surmise, conjecture and conclusory allegations are not enough; plaintiff must make an affirmative
Trump_Tower
Disorder of depression and mood episodes
possible mechanisms connecting cytokine levels to mood changes. They conjecture that immune dysfunction is relevant to some but not all bipolar disorder
Bipolar_disorder
Regions south of the Sahara
40.8% of the total. These numbers are estimates and remain a matter of conjecture. See Amadu Jacky Kaba. The spread of Christianity and Islam in Africa:
Sub-Saharan_Africa
Castilian warlord and Prince of Valencia from 1094 to 1099
which the Christians may have transliterated and adopted. It has been conjectured that he received the honorific title and respectful treatment of contemporaries
El_Cid
English singer (born 1974)
about his sexuality. He also said he is "sad" about the related tabloid conjecture, yet does not fear being perceived as homosexual: "You want to be an ally
Robbie_Williams
Murder of 12-year-old Chinese-Spanish child in 2013
failed to clarify satisfactorily, giving rise to numerous hypotheses and conjectures with little real basis in the press and social networks. On 16 March
Murder_of_Asunta_Basterra
American SeaWorld trainer (1969–2010)
had targeted Brancheau because she had worn her hair in a ponytail was conjecture and that "there had to be more to this story". Her family has stated they
Dawn_Brancheau
1982 American fantasy film by John Milius
breaking of his father's sword—"[fulfills] a snickering spectrum of Oedipal conjecture" and asserts Homer's view that "the sword does not make the hero, but
Conan the Barbarian (1982 film)
Conan_the_Barbarian_(1982_film)
their script instead wrote their native Canaanite language. It has been conjectured that the community selected a small number of the hieroglyphs commonly
History_of_the_alphabet
American television host and writer
local television stations in the United States and later for CBS News and ABC News, the former of which earned O'Reilly two Emmy Awards and two National
Bill O'Reilly (political commentator)
Bill_O'Reilly_(political_commentator)
King of the United Kingdom from 1901 to 1910
humanitarian Agnes Keyser; and Alice Keppel. At least fifty-five liaisons are conjectured. How far these relationships went is not always clear. Edward always
Edward_VII
ABC CONJECTURE
ABC CONJECTURE
Female
Gaelic
Variant form of Irish Gaelic Meadhbh, MÉABH means "intoxicating." In mythology, this is the name of a warrior queen of Connacht, the wife of Ailill.
Male
English
Short form of Hebrew Abraham, ABE means "father of a multitude,"Â and other names beginning with Ab-.
Male
Hebrew
Pet form of Hebrew Abraham, ABI means "father of a multitude." Also spelled Avi. Compare with feminine Abi.
Boy/Male
Indian
Servant, Devotee, Slave
Boy/Male
Biblical
My father.
Boy/Male
Scottish
Bank.
Boy/Male
Australian, British, English, French, German
Father
Male
Portuguese
Portuguese name ABÃLIO means "able; proficient; skillful."
Female
Irish
Variant form of Irish Meadhbh, MÉABH means "intoxicating." In mythology, this is the name of a warrior queen of Connacht, the wife of Ailill.
Boy/Male
African, German, Gujarati, Hindu, Indian, Malayalam, Tamil, Telugu, Turkish
My Father; All Given to Good
Boy/Male
Hebrew American
Father of a multitude. In Genesis the Old Testament patriarch Abram's name was changed to Abraham...
Boy/Male
American, Australian, Chinese, Christian, Dutch, Hebrew
Father of a Multitude; Breath; He who is High is Father; Diminutive of Abraham
Boy/Male
Arabic, Muslim
Name of a Great Jurist and Pupil of Imam Abu Hanifah
Boy/Male
Arabic, Islamic
Austere; Frowning
Boy/Male
Arabic, Muslim
Well-known Sahabi Abu Moosa Al-ashari
Male
French
French form of German Abelard, ABÉLARD means "noble strength."
Female
English
Anglicized form of Hebrew Abiy, ABI means "my father." In the bible, this is the name of the mother of King Hezekiah. Compare with masculine Abi.Â
Boy/Male
Muslim
Servant, Devotee, Slave
Girl/Female
German
Highborn and Steadfast
Boy/Male
African, Arabic, Australian, French, German
Father (Pahlavi); Master
ABC CONJECTURE
ABC CONJECTURE
Surname or Lastname
English (Wiltshire)
English (Wiltshire) : topographic name for someone who lived by a well dedicated to St. Bride or by a stream frequented by birds (Old English bridd).
Female
Hebrew
(×”Öµ× Ö°×™Ö¸×”) Variant spelling of Hebrew Chenya, HENYA means "grace of the Lord."
Girl/Female
Muslim/Islamic
Success height
Boy/Male
Indian, Sanskrit
The Creator
Boy/Male
Biblical American Hebrew
My messenger, my angel'.
Boy/Male
Muslim
Ringneck dove
Boy/Male
Arabic, Muslim, Pashtun
Little Heart
Boy/Male
Tamil
Abhinava | அபிநவாÂ
Young, New, Novel, Innovative, Quite new, Fresh, Modern, A sakta notable for his great leaning and spiritual attainment
Boy/Male
Indian, Sanskrit
Beauty; Movable
Girl/Female
Indian, Sikh
Lotus in the Lamp of Light
ABC CONJECTURE
ABC CONJECTURE
ABC CONJECTURE
ABC CONJECTURE
ABC CONJECTURE
n.
An arch.
n.
A ferryboat. See Bac, 1.
n.
The fifth month of the Jewish year according to the ecclesiastical reckoning, the eleventh by the civil computation, coinciding nearly with August.
n.
A part of a circle comprehended between two radii and the included arc.
n.
A curvature in the shape of a circular arc or an arch; as, the colored arc (the rainbow); the arc of Hadley's quadrant.
n.
A broad, flatbottomed ferryboat, usually worked by a rope.
n.
A vat or cistern. See 1st Back.
n.
The apparent arc described, above or below the horizon, by the sun or other celestial body. The diurnal arc is described during the daytime, the nocturnal arc during the night.
n.
Among weavers, yarn for the warp. Hence, abb wool is wool for the abb.
n.
The eighth part of a circle; an arc of 45 degrees.
a.
Of or pertaining to an arc.
n.
A portion of a curved line; as, the arc of a circle or of an ellipse.
v. t.
The number of degrees which, if added to a specified arc, make it 180¡; the quantity by which an arc or an angle falls short of 180 degrees, or an arc falls short of a semicircle.
n.
An arc of a circle which is equal to the radius, or the angle measured by such an arc.
v. t. & i.
Alt. of Abye
n.
The tangent of half an arc.