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Concept in algebraic number theory
In number theory, Heegner numbers are square-free positive integers d {\displaystyle d} such that the imaginary quadratic field Q ( − d ) {\displaystyle
Heegner_number
German mathematician
as Heegner's.) The result is now often called the Stark–Heegner theorem. List of amateur mathematicians Stark–Heegner theorem Heegner number Heegner point
Kurt_Heegner
Natural number
natural number following 66 and preceding 68. 67 is the 19th prime number, a Chen prime, an irregular prime, a lucky prime, a Heegner number, a super-prime
67_(number)
Topics referred to by the same term
quadratic fields of class number 1. A Heegner number is a number n such that Q(√−n) is an imaginary quadratic field of class number 1. This disambiguation
Heegner
Natural number
a Chen prime. 43 is also a Wagstaff prime, a Gaussian prime, and a Heegner number. 43 is the fourth term of Sylvester's sequence. 43 is the largest prime
43_(number)
Natural number
\mathrm {J_{1}} } and J 3 {\displaystyle \mathrm {J_{3}} } . 19 is a Heegner number. The James Webb Space Telescope's optical elements are in an array of
19_(number)
Special point on a modular curve in mathematics
and named after Kurt Heegner, who used similar ideas to prove Gauss's conjecture on imaginary quadratic fields of class number one. The Gross–Zagier
Heegner_point
Natural number
quadratic field of class number two, where 163 is the largest such (Heegner) number of class one. The twenty-first prime number 73 is the largest member
21_(number)
Listing all imaginary quadratic fields with a given class number
Stark and Bryan Birch (e.g. on the Stark–Heegner theorem and Heegner number) was the position clarified and Heegner's work understood. Practically simultaneously
Class_number_problem
Quadratic imaginary number fields with unique factorisation
In number theory, the Heegner theorem or Stark-Heegner theorem establishes the complete list of the quadratic imaginary number fields whose rings of integers
Stark–Heegner_theorem
Natural number
5 2 {\displaystyle 2^{5}-5^{2}} ), and the fourth Heegner number. Seven is the lowest natural number that cannot be represented as the sum of the squares
7
Number divisible only by 1 and itself
explanation for this phenomenon led to the deep algebraic number theory of Heegner numbers and the class number problem. The Hardy–Littlewood conjecture F predicts
Prime_number
Natural number
1 + − 163 ) / 2 {\displaystyle n=(-1+{\sqrt {-163}})/2} . 163 is a Heegner number, the largest of the nine such numbers. That is, the ring of integers
163_(number)
Fast method for calculating the digits of π
the algorithm on y-cruncher. The algorithm is based on the negated Heegner number d = − 163 {\displaystyle d=-163} , the j-function j ( 1 + i 163 2 )
Chudnovsky_algorithm
Negative integer two units from the origin in mathematics
domain. According to the Stark–Heegner theorem, only nine negative numbers have this property, corresponding to Heegner numbers. Negative two also makes
−2
Natural number
cube-sum number greater than one. A number that is 4 or 5 modulo 9 cannot be represented as the sum of three cubes. There are nine Heegner numbers, or
9
Formula whose values are the prime numbers
which is also implicitly quadratic, and the class number; this polynomial is related to the Heegner number 163 = 4 ⋅ 41 − 1 {\displaystyle 163=4\cdot 41-1}
Formula_for_primes
Field (mathematics) generated by the square root of an integer
Eisenstein–Kronecker number Genus character Heegner number Infrastructure (number theory) Quadratic integer Quadratic irrational Stark–Heegner theorem Dedekind
Quadratic_field
Theorem Class number problem for imaginary quadratic fields Stark–Heegner theorem Heegner number Langlands program Different ideal Dedekind domain Splitting
List of algebraic number theory topics
List_of_algebraic_number_theory_topics
Criterion for existence of polynomial roots
In mathematics, Heegner's lemma is a lemma used by Kurt Heegner in his paper on the class number problem. His lemma states that if y 2 = a 4 x 4 + a 3
Heegner's_lemma
Mathematical concept
to 2 modulo 3. Heegner number List of topics named after Leonhard Euler Formula for primes Ulam spiral Weisstein, Eric W. "Lucky Number of Euler". mathworld
Lucky_numbers_of_Euler
Natural number
imaginary quadratic fields with class number of 7, negated (see also Heegner numbers). 71 is the largest number which occurs as a prime factor of an order
71_(number)
Constant e raised to the power of pi
6403203 + 744. This is an application of Heegner numbers, where 163 is the Heegner number in question. This number was discovered in 1859 by the mathematician
Gelfond's_constant
Area of a right triangle with rational-numbered sides
1073/pnas.1216991109. PMC 3535615. PMID 23213259. Paul Monsky (1990), "Mock Heegner Points and Congruent Numbers", Mathematische Zeitschrift, 204 (1): 45–67
Congruent_number
Coincidence in mathematics
is the case for most). It is a consequence of the fact that 163 is a Heegner number. There are several integers k = 2198 , 422151 , 614552 , 2508952 , 6635624
Mathematical_coincidence
Natural number
fundamental Coxeter groups in six-dimensional space. The sum of the first eight Heegner numbers is 153. The Gospel of John (chapter 21:1–14) includes the miraculous
153_(number)
Natural number
rods per mile. a number connected with both Klein's J-invariant and the Heegner numbers. Specifically: 5280 = − j ( 1 2 ( 1 + i 67 ) ) 3 . {\displaystyle
5000_(number)
British mathematician (born 1931)
formulated ideas on the role of Heegner points (he was one of those reconsidering Kurt Heegner's original work on the class number one problem, which had not
Bryan_John_Birch
Conjecture on zeros of the zeta function
Gauss's list of imaginary quadratic fields with class number 1 is complete, though Baker, Stark and Heegner later gave unconditional proofs of this without
Riemann_hypothesis
English mathematician (1939–2018)
known for his work on effective methods in number theory, in particular those arising from transcendental number theory. Alan Baker was born into a Jewish
Alan_Baker_(mathematician)
In number theory, measure of non-unique factorization
and proven by Kurt Heegner, although Heegner's proof was not believed until Harold Stark gave a later proof in 1967 (see Stark–Heegner theorem). This is
Ideal_class_group
German mathematician
results is an alternative solution to the class number 1 problem, building on the original Stark–Heegner theorem. Die Berechnung der Klassenzahl abelscher
Curt_Meyer
American mathematician (1950–2025)
Gross–Zagier formula in 1986 has been influential in modern number theory. It describes the height of Heegner points in terms of a derivative of the L-function
Benedict_Gross
Chinese-American mathematician (born 1962)
totally real fields through his work relating the Néron–Tate height of Heegner points to special values of L-functions in (Zhang 1997, 2001). In 2013
Shou-Wu_Zhang
Mathematical concept
indexed by fields. They were introduced by Kolyvagin (1990) in his work on Heegner points on modular elliptic curves, which was motivated by his earlier paper
Euler_system
American mathematician (born 1939)
problem, in effect correcting and completing the earlier work of Kurt Heegner, and for Stark's conjecture. More recently, he collaborated with Audrey
Harold_Stark
American mathematician
L-series of an elliptic curve evaluated at 1 to the height of a certain Heegner point. This theorem has some applications, including implying cases of
Don_Zagier
Unproved conjecture in mathematics
p. 462. Zhang, Wei (2013). "The Birch–Swinnerton-Dyer conjecture and Heegner points: a survey". Current Developments in Mathematics. 2013: 169–203.
Birch and Swinnerton-Dyer conjecture
Birch_and_Swinnerton-Dyer_conjecture
Indian mathematician (1887–1920)
{58}}}=396^{4}-104.000000177\dots .} This might be compared to Heegner numbers, which have class number 1 and yield similar formulae. Ramanujan's series for π
Srinivasa_Ramanujan
Chinese mathematician (born 1981)
180 (2014), No. 3, 971–1049. "Selmer groups and the indivisibility of Heegner points" Archived 12 August 2017 at the Wayback Machine, Cambridge Journal
Wei_Zhang_(mathematician)
Root of a quadratic polynomial with a unit leading coefficient
and proven by Kurt Heegner, although Heegner's proof was not believed until Harold Stark gave a later proof in 1967 (see Stark–Heegner theorem). This is
Quadratic_integer
Concept in number theory
Siegel, Carl, Advanced Analytic Number Theory Bertolini, Massimo; Darmon, Henri (2009), "The rationality of Stark-Heegner points over genus fields of real
Genus_character
Topics referred to by the same term
Antarctica Stark Round Barn, near Unityville, South Dakota Stark–Heegner theorem, in number theory zk-STARK (zero-knowledge Scalable Transparent Argument
Stark
Canadian mathematician
International Congress of Mathematicians in Madrid. Uniform distribution of Heegner Points, Inventiones Mathematicae, Vol. 148, 2002, pp. 1–48 (Proof of a
Vinayak_Vatsal
Prize awarded by the American Mathematical Society
doi:10.1090/S0273-0979-1985-15352-2. Gross, Benedict; Zagier, Don (1986). "Heegner points and derivatives of L-series" (PDF). Inventiones Mathematicae. 84
Cole_Prize
American mathematician at Duke University
abelian varieties, and units in number fields. In particular, Dasgupta's research has focused on the Stark conjectures and Heegner points. In 2009, Dasgupta
Samit_Dasgupta
Compact Riemann surface of genus 3
theory, Fermat's Last Theorem, and the Stark–Heegner theorem on imaginary quadratic number fields of class number one; see (Levy 1999) for a survey of properties
Klein_quartic
Any number that is not an integer but is very close to one
occurrences of non-coincidental near-integers involve the three largest Heegner numbers: e π 43 ≈ 8847 36743.99977 7466 {\displaystyle e^{\pi {\sqrt {43}}}\approx
Almost_integer
Theory of a class of elliptic curves
if K has class number one, then j(a) = j(O) is a rational integer: for example, j(Z[i]) = j(i) = 1728. Algebraic Hecke character Heegner point Hilbert's
Complex_multiplication
Mathematical conjecture about zeros of L-functions
Gauss's list of imaginary quadratic fields with class number 1 is complete, though Baker, Stark and Heegner later gave unconditional proofs of this without
Generalized Riemann hypothesis
Generalized_Riemann_hypothesis
Chinese mathematician
MR 2782253, S2CID 122139021. Tian, Ye (2014), "Congruent numbers and Heegner points", Cambridge Journal of Mathematics, 2 (1): 117–161, arXiv:1210.8231
Tian_Ye_(mathematician)
and Thomas in 1974. Class numbers of imaginary quadratic fields. In 1952 Heegner published a solution to this problem. His paper was not accepted as a complete
List_of_incomplete_proofs
equation (number theory) Sophie Germain's theorem (number theory) Sphere packing theorems in dimensions 8 and 24 (geometry, modular forms) Stark–Heegner theorem
List_of_theorems
Type of integral domain
{Q} [{\sqrt {-d}}]} will fail to be a UFD unless d is one of the nine Heegner numbers. The ring of formal power series over the complex numbers is a
Unique_factorization_domain
Varying methods used to calculate pi
needed] because there are only finitely many Heegner numbers and negative discriminants d with class number h(−d) = 1, and d = 163 is the largest one in
Approximations_of_pi
Algebra with unique prime factorization
(Gauss's conjecture was proven more than one hundred years later by Kurt Heegner, Alan Baker and Harold Stark.) However, this was understood (only) in the
Dedekind_domain
Bronze medal awarded by the Royal Society (London)
curves, through the Birch-Swinnerton-Dyer conjecture and the theory of Heegner points." 2021 Frances Kirwan British "For her research on quotients in
Sylvester_Medal
Romanian-Canadian mathematician
Iovita, Adrian; Spiess, Michael (2003). "Derivatives of p-adic L-functions, Heegner cycles and monodromy modules attached to modular forms". Inventiones Mathematicae
Adrian_Ioviță
Mathematical function
Maass form to E . The simultaneous generating series for the values on Heegner divisors and integrals along geodesic cycles of Klein's J-function (normalized
Harmonic_Maass_form
German mathematician and historian of mathematics
Mathematical Institute in Göttingen), as well as historical research on Kurt Heegner, Bartel Leendert van der Waerden, Diophantus of Alexandria, and Leonhard
Norbert_Schappacher
British mathematician
collaborated with Norbert Schappacher on elucidating the biography of Kurt Heegner. In 1984 Patterson received the Whitehead Prize of the London Mathematical
Samuel_James_Patterson
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