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In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer d such that the imaginary quadratic field [] has class number 1. Equivalently, the ring of algebraic integers of Q [ − d ] {\displaystyle \mathbb {Q} \left[{\sqrt {-d}}\right]} has unique factorization .
Kurt Heegner (German: [ˈheːɡnɐ]; 16 December 1893 – 2 February 1965) was a German private scholar from Berlin, who specialized in radio engineering and mathematics. He is famous for his mathematical discoveries in number theory and, in particular, the Stark–Heegner theorem .
Heegner points are special points on elliptic curves The Stark–Heegner theorem identifies the imaginary 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.
Let Q denote the set of rational numbers, and let d be a square-free integer. The field Q(√ d) is a quadratic extension of Q. The class number of Q(√ d) is one if and only if the ring of integers of Q(√ d) is a principal ideal domain. The Baker–Heegner–Stark theorem can then be stated as follows:
The number e π √ 163 is known as Ramanujan's constant. Its decimal expansion is given by: e π √ 163 = 262 537 412 640 768 743.999 999 999 999 250 072 59... (sequence A060295 in the OEIS) which suprisingly turns out to be very close to the integer 640320 3 + 744: This is an application of Heegner numbers, where 163 is the
19 is the sixth Heegner number. [50] 67 and 163, respectively the 19th and 38th prime numbers, are the two largest Heegner numbers, of nine total. The sum of the first six Heegner numbers 1, 2, 3, 7, 11, and 19 sum to the seventh member and fourteenth prime number, 43. All of these numbers are prime, aside from the unit.
The phenomenon is related to the Ulam spiral, which is also implicitly quadratic, and the class number; this polynomial is related to the Heegner number =. There are analogous polynomials for p = 2 , 3 , 5 , 11 and 17 {\displaystyle p=2,3,5,11{\text{ and }}17} (the lucky numbers of Euler ), corresponding to other Heegner numbers.
In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. [1] Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.