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There is a prime between any two square numbers
Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between n 2 {\displaystyle n^{2}} and ( n + 1 ) 2 {\displaystyle
Legendre's_conjecture
Mathematical conjecture
The conjecture is named after Henri Brocard. It is widely believed that this conjecture is true. However, it remains unproven as of 2025. Legendre's conjecture
Brocard's_conjecture
Four basic unsolved problems about prime numbers
conjecture: Are there infinitely many primes p such that p + 2 is prime? Legendre's conjecture: Does there always exist at least one prime between consecutive
Landau's_problems
French mathematician (1752–1833)
theory. His 1798 conjecture of the prime number theorem was rigorously proved by Hadamard and de la Vallée-Poussin in 1896. Legendre did an impressive
Adrien-Marie_Legendre
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
Existence of a prime number between each square and pronic number
Oppermann's conjecture is an unsolved problem in mathematics on the distribution of prime numbers. It is closely related to but stronger than Legendre's conjecture
Oppermann's_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
Conjecture about gaps between prime numbers
Andrica's conjecture (named after Romanian mathematician Dorin Andrica [es]) is a conjecture regarding the gaps between prime numbers. The conjecture states
Andrica's_conjecture
2000, six remain unsolved to date: Birch and Swinnerton-Dyer conjecture Hodge conjecture Navier–Stokes existence and smoothness[when?] P versus NP Riemann
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Estimation in number theory
log e ( x ) {\displaystyle \log _{e}(x)} . In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an
Cramér's_conjecture
German mathematician
to Legendre and Steinitz) on projectively unique polyhedra." In joint work with June Huh and Eric Katz, he resolved the Heron–Rota–Welsh conjecture on
Karim_Adiprasito
Difference between two successive prime numbers
1)2 for n sufficiently large (see Legendre's conjecture). To verify this, a stronger result such as Cramér's conjecture would be needed. Huxley in 1972
Prime_gap
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
Result on density of prime numbers
values of n {\displaystyle n} . The similar and still unsolved Legendre's conjecture asks whether for every n ≥ 1 {\displaystyle n\geq 1} , there is
Bertrand's_postulate
condition Legendre–Fenchel transformation Legendre's conjecture Legendre's constant Legendre's differential equation Legendre's equation Legendre's formula
List of things named after Adrien-Marie Legendre
List_of_things_named_after_Adrien-Marie_Legendre
Number divisible only by 1 and itself
. {\displaystyle 2k.} Andrica's conjecture, Brocard's conjecture, Legendre's conjecture, and Oppermann's conjecture all suggest that the largest gaps
Prime_number
Prime-generating mathematical constant
with this middle exponent to always produce primes. Moreover, if Legendre's conjecture is true, the middle exponent can be replaced with value 2 (sequence
Mills'_constant
Bound on the gaps between prime numbers
In number theory, Firoozbakht's conjecture (or the Firoozbakht conjecture) is a conjecture about the distribution of prime numbers. It is named after the
Firoozbakht's_conjecture
conjecture is a conjecture, named after Fedor Bogomolov, in arithmetic geometry about algebraic curves that generalizes the Manin–Mumford conjecture in
Bogomolov_conjecture
is the sum of a fixed number of kth powers. 1796 — Adrien-Marie Legendre conjectures the prime number theorem. 1801 — Disquisitiones Arithmeticae, Carl
Timeline_of_number_theory
Visualization of the prime numbers
a high asymptotic density of them, although there is a well-supported conjecture as to what that asymptotic density should be. In 1932, 31 years prior
Ulam_spiral
Characterization of how many integers are prime
Based on the tables by Anton Felkel and Jurij Vega, Adrien-Marie Legendre conjectured in 1797 or 1798 that π(a) is approximated by the function a / (A
Prime_number_theorem
Used to count, measure, and label
to describe the method of trial division. In 1796, Adrien-Marie Legendre conjectured the prime number theorem, describing the asymptotic distribution
Number
Chinese number theorist
Luogeng. His work on the twin prime conjecture, Waring's problem, Goldbach's conjecture and Legendre's conjecture led to progress in analytic number theory
Chen_Jingrun
Exploring properties of the integers with complex analysis
the asymptotic law of distribution of prime numbers. Adrien-Marie Legendre conjectured in 1797 or 1798 that π(a) is approximated by the function a/(A ln(a) + B)
Analytic_number_theory
Mathematical conjecture on the Riemann zeta function
In mathematics, the Lindelöf hypothesis is a conjecture by Finnish mathematician Ernst Leonard Lindelöf about the rate of growth of the Riemann zeta function
Lindelöf_hypothesis
constructed using only a compass and straightedge. 1796—Adrien-Marie Legendre conjectures the prime number theorem. 1797—Caspar Wessel associates vectors with
Timeline_of_mathematics
Number whose sums of distinct divisors represent all smaller numbers
for any positive real x {\displaystyle x} , a result analogous to Legendre's conjecture for primes. Moreover, for all sufficiently large x {\displaystyle
Practical_number
Natural number
express the asymptotic behavior of the prime-counting function. The Weil's conjecture on Tamagawa numbers states that the Tamagawa number τ ( G ) {\displaystyle
1
French mathematician (1906-1998)
Weil conjecture, around 1967, which later under pressure from Serge Lang (resp. of Jean-Pierre Serre) became known as the Taniyama–Shimura conjecture (resp
André_Weil
of polynomials with coefficients in finite fields. Adrien-Marie Legendre conjectures the prime number theorem. May 14 – Edward Jenner administers the
1796_in_science
American mathematician
solved a conjecture of Antoni Zygmund, establishing mean convergence of the partial sums in L p {\displaystyle L^{p}} norms for the Legendre polynomials
Harry_Pollard_(mathematician)
Constant of proportionality of prime number density
{\displaystyle \log _{e}(x)} . Legendre's constant is a mathematical constant occurring in a formula constructed by Adrien-Marie Legendre to approximate the behavior
Legendre's_constant
Field of knowledge
across mathematics. A prominent example is Fermat's Last Theorem. This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994
Mathematics
On the divisibility of solutions to Fermat's Last Theorem for prime exponent
which the violation of the Fermat Theorem would occur and most likely the conjecture is true that for given n {\displaystyle n} the auxiliary prime may be
Sophie_Germain's_theorem
Type of prime number conjectured to exist
Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known. Let p {\displaystyle p} be a prime
Wall–Sun–Sun_prime
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
Theorem on the number of primes in arithmetic sequences
cyclotomic polynomials. The general form of the theorem was first conjectured by Legendre in his attempted unsuccessful proofs of quadratic reciprocity —
Dirichlet's theorem on arithmetic progressions
Dirichlet's_theorem_on_arithmetic_progressions
Norwegian mathematician (1885–1978)
on Legendre's version of the sieve of Eratosthenes, now known as the Brun sieve, which addresses additive problems such as Goldbach's conjecture and
Viggo_Brun
Polynomial whose coefficients are all 1 or −1
{n+1}}} at every point of the circle. Littlewood conjectured that this was possible; the conjecture was proved in 2020 by Paul Balister, Béla Bollobás
Littlewood_polynomial
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
Every natural number can be represented as the sum of four integer squares
Lagrange's four-square theorem, also known as Bachet's conjecture, states that every nonnegative integer can be represented as a sum of four non-negative
Lagrange's four-square theorem
Lagrange's_four-square_theorem
Open conjecture that no real circulant Hadamard matrix has order greater than 4
and combinatorics, Ryser's conjecture on circulant Hadamard matrices, also called the circulant Hadamard matrix conjecture, states that no real circulant
Ryser's conjecture on circulant Hadamard matrices
Ryser's_conjecture_on_circulant_Hadamard_matrices
List of mathematical contexts in which exponentiated terms are summed
|x/a|^{k}+|y/b|^{k}=1} . The squircle is the case k = 4, a = b. Euler's sum of powers conjecture (disproved) concerns situations in which the sum of n integers, each a
Sums_of_powers
Sequence of digital values used for synchronisation
known Barker sequences have lengths 2, 3, 4, 5, 7, 11 and 13. It is conjectured that no longer sequence exists; this is proved for odd lengths, while
Barker_code
Italian-born American mathematician (1923–2023)
and the result became known as the Calabi conjecture. In 1957, Calabi published a paper in which the conjecture was stated as a proposition, but with an
Eugenio_Calabi
Three results related to the density of prime numbers
infinity!); Legendre's argument is heuristic; and Chebyshev's proof, although perfectly sound, makes use of the Legendre-Gauss conjecture, which was not
Mertens'_theorems
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
List_of_incomplete_proofs
Chinese-American mathematician (born 1949)
recognition of his contributions to partial differential equations, the Calabi conjecture, the positive energy theorem, and the Monge–Ampère equation. Yau is considered
Shing-Tung_Yau
Gives conditions for the solvability of quadratic equations modulo prime numbers
reciprocity. The quadratic reciprocity theorem was conjectured by Leonhard Euler and Adrien-Marie Legendre and first proved by Carl Friedrich Gauss, who referred
Quadratic_reciprocity
Israeli mathematician and professor
duality, Legendre and Fourier transform from axiomatic viewpoint (with V. Milman) and discovery of an astonishing link between Mahler's conjecture in convexity
Shiri_Artstein
Traditional board game for two players
has even less documentation than its migration west, making it largely conjectured. The word xiàngqí (象棋) was used in China to refer to a game from 569
Chess
in each of the 50 rows. (sequence A000040 in the OEIS). The Goldbach conjecture verification project reports that it has computed all primes smaller than
List_of_prime_numbers
Triangular decomposition Sturm's theorem Descartes' rule of signs Carlitz–Wan conjecture Polynomial decomposition, factorization under functional composition Delta
List_of_polynomial_topics
Set of distances defined from a set of points
a large distance set (for varying definitions of "large"): Falconer's conjecture is the statement that, for a collection of points in d {\displaystyle
Distance_set
Ways to estimate the size of sifted sets of integers
These can be considered to be near-misses to the twin prime conjecture and the Goldbach conjecture respectively. The fundamental lemma of sieve theory, which
Sieve_theory
Surface that locally minimizes its area
the positive mass conjecture, the Penrose conjecture) and three-manifold geometry (e.g. the Smith conjecture, the Poincaré conjecture, the Thurston Geometrization
Minimal_surface
Number, approximately 3.14
von Lindemann proved that π is transcendental, confirming a conjecture made by both Legendre and Euler. The first recorded use of the symbol π in circle
Pi
Constructions of Hadamard matrices
ATA = I. This appears to be the first published statement of the Hadamard conjecture. A matrix of size 92 was eventually constructed by Baumert, Golomb, and
Paley_construction
Infinitely many prime numbers exist
verified his statement for all numbers in the interval [2, 3 × 106]. His conjecture was completely proved by Chebyshev (1821–1894) in 1852 and so the postulate
Euclid's_theorem
Study of collection and analysis of data
Institute 5(4): 321–328. JSTOR 1400906 Franklin, James (2002). The Science of Conjecture. Baltimore: Taylor & Francis. ISBN 978-0-8018-7109-2. Grattan-Guinness
Statistics
Sum of the inverses of the positive integers cubed is irrational
{\displaystyle 2n+1} ( n > 1 {\displaystyle n>1} ) (though they are conjectured to be irrational). Leonhard Euler proved that if n is a positive integer
Apéry's_theorem
Four integers where the sum of the squares of three equals the square of the fourth
quadruples in which all entries are less than 30. Beal conjecture Euler brick Euler's sum of powers conjecture Euler-Rodrigues formula for 3D rotations Fermat
Pythagorean_quadruple
1798 textbook by Carl Friedrich Gauss
published, number theory consisted of a collection of isolated theorems and conjectures. Gauss brought the work of his predecessors together with his own original
Disquisitiones_Arithmeticae
Relation between sides of a right triangle
question is why Euclid did not use this proof, but invented another. One conjecture is that the proof by similar triangles involved a theory of proportions
Pythagorean_theorem
Mathematical law, a generalization of quadratic reciprocity
without the use of roots of unity. The Langlands program includes several conjectures for general reductive algebraic groups, which for the special of the
Reciprocity_law
Numerical ordinary differential equations Bendixson–Dulac theorem Gradient conjecture Recurrence plot Limit cycle Initial value problem Clairaut's equation
List of dynamical systems and differential equations topics
List_of_dynamical_systems_and_differential_equations_topics
the Mordell conjecture (a conjecture dating back to 1922). Other theorems proved in this paper include an instance of the Tate conjecture (relating the
List of publications in mathematics
List_of_publications_in_mathematics
Methods to test or prove primality
accepts this conjecture then the Goldwasser–Kilian algorithm terminates in expected polynomial time for every input. Now consider another conjecture, which
Elliptic_curve_primality
Continuous functional calculus Borel functional calculus Hilbert–Pólya conjecture Lp space Hardy space Sobolev space Tsirelson space ba space Uniform norm
List of functional analysis topics
List_of_functional_analysis_topics
complete specimen have been found. Their body mass, especially, is largely conjecture because soft tissue was rarely fossilized. Generally, the size of extinct
Largest_prehistoric_animals
German polymath and scholar (1777–1855)
his arguments, proved the central conjecture, and remarked that this theorem is equivalent to the Kepler conjecture for regular arrangements. In two papers
Carl_Friedrich_Gauss
Branch of pure mathematics
of Pell's equations. Adrien-Marie Legendre (1752–1833) stated the law of quadratic reciprocity. He also conjectured what amounts to the prime number theorem
Number_theory
Conditions for switching order of integration in calculus
to bounded measurable functions on a product of intervals. Beppo Levi conjectured that the theorem could be extended to functions that are integrable rather
Fubini's_theorem
coordinates are non-negative for points in the convex hull. Borsuk's conjecture - a conjecture about the number of pieces required to cover a body with a larger
List_of_convexity_topics
Computation modulo a fixed integer
Rosetta Code, modular arithmetic was used to disprove Euler's sum of powers conjecture on a Sinclair QL microcomputer using just one-fourth of the integer precision
Modular_arithmetic
Number measuring the chance an event occurs
Hacking's The Emergence of Probability and James Franklin's The Science of Conjecture for histories of the early development of the very concept of mathematical
Probability
Integer that is a perfect square modulo some integer
Euler, Lagrange, Legendre, and other number theorists of the 17th and 18th centuries established theorems and formed conjectures about quadratic residues
Quadratic_residue
Mathematical concept in prime numbers
idoneal numbers found by Leonhard Euler and Carl Friedrich Gauss and conjectured to be the only such numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13
Idoneal_number
used to prove the Poincaré conjecture Tzitzeica equation Lorenz equations Rabinovich–Fabrikant equations General Legendre equation Heat equation Ishimori
List of named differential equations
List_of_named_differential_equations
2008 mathematics book
theorem, Betti numbers, and Grigori Perelman's proof of the Poincaré conjecture. An appendix includes instructions for creating paper and soap-bubble
Euler's_Gem
Hungarian mathematician
Knapowski he proved results concerning Chebyshev's bias. The Erdős–Turán conjecture makes a statement about primes in arithmetic progression. Much of Turán's
Pál_Turán
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 common
Factorial
French mathematician, physicist, and philosopher (1776–1831)
and mathematicians, Germain's name was not among them; H. J. Mozans conjectured that despite the salience of her work to the tower's construction, she
Sophie_Germain
Numbers obtained by adding the two previous ones
The only triangular Fibonacci numbers are 1, 3, 21, and 55, which was conjectured by Vern Hoggatt and proved by Luo Ming. No Fibonacci number can be a
Fibonacci_sequence
French mathematician (1811–1832)
might have prompted him to provoke the duel himself on her behalf. This conjecture is also supported by other letters Galois later wrote to his friends the
Évariste_Galois
Mathematical function
it was the first known scattering amplitude in string theory, first conjectured by Gabriele Veneziano. It also occurs in the theory of the preferential
Beta_function
Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n
Chu (2004). "A Binomial Coefficient Identity Associated with Beukers' Conjecture on Apery Numbers" (PDF). The Electronic Journal of Combinatorics. 11 N15
Harmonic_number
Family of power series in mathematics
{1}{2}},s;c;x)^{2}} was used by de Branges to prove the Bieberbach conjecture. Many of the special functions in mathematics are special cases of the
Generalized hypergeometric function
Generalized_hypergeometric_function
Number that is not a ratio of integers
because e and π are not known to be algebraically independent. Schanuel's conjecture would imply that all of the above numbers are irrational and even transcendental
Irrational_number
Type of non-Euclidean geometry
experience does not necessarily rule out other geometries. The geometrization conjecture gives a complete list of eight possibilities for the fundamental geometry
Hyperbolic_geometry
Primality test for Fermat numbers
before any more Pépin tests can be run in a reasonable amount of time. Conjecture 4. Fermat primes are finite - Pepin tests story, according to Leonid Durman
Pépin's_test
Mathematical functions
terms can be multiplied, as a consequence of uniform convergence. Gauss conjectured that ln N ( ϖ ) = π / 2 {\displaystyle \ln N(\varpi )=\pi /2} (this
Lemniscate_elliptic_functions
Positive real number which when multiplied by itself gives 5
Martin (July 18, 2013), Markov's Theorem and 100 Years of the Uniqueness Conjecture: A Mathematical Journey from Irrational Numbers to Perfect Matchings,
Square_root_of_5
Historical development of geometry
is not certain what practical use these arithmetic rules had. The best conjecture is that they were part of religious ritual. A Hindu home was required
History_of_geometry
Distance along a portion of a meridian, for use in geodesy
of 1/150 considered as unacceptable. This value was the result of a conjecture based on too limited data. Another flattening of the Earth was calculated
Meridian_arc
and its Applications to Number Theory Néron model Birch–Swinnerton–Dyer conjecture Moduli space for abelian varieties Duality of abelian varieties c.1967
Timeline_of_abelian_varieties
Retrieved 14 July 2026. "ChatGPT just proved another 50-year-old math conjecture". scientificamerican.com. Retrieved 18 June 2026. "DeepMind CEO calls
Timeline of artificial intelligence
Timeline_of_artificial_intelligence
Function used in local class field theory related to reciprocity laws
{\displaystyle K_{2}^{M}(K)/2} . This is the first step towards the Milnor conjecture. The Hilbert symbol can also be used to denote the central simple algebra
Hilbert_symbol
Conditions in number theory
on biquadratic reciprocity. In the first one (1828) he proved Euler's conjecture about the biquadratic character of 2. In the second one (1832) he stated
Quartic_reciprocity
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LEGENDRES CONJECTURE
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LEGENDRES CONJECTURE
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