Search results
Results From The WOW.Com Content Network
Euler's prime-generating polynomial + +, which gives (distinct) primes for n = 0, ..., 39, is related to the Heegner number 163 = 4 · 41 − 1.. Rabinowitz [3] proved that + + gives primes for =, …, if and only if this quadratic's discriminant is the negative of a Heegner number.
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.
The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. The latter is the function in the definition. They also occur in combinatorics , specifically when counting the number of alternating permutations of a set with an even number of elements.
Kurt Heegner was a German mathematician; 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.
Leonhard Euler published the polynomial k 2 − k + 41 which produces prime numbers for all integer values of k from 1 to 40. Only 6 lucky numbers of Euler exist, namely 2, 3, 5, 11, 17 and 41 (sequence A014556 in the OEIS). [1] Note that these numbers are all prime numbers. The primes of the form k 2 − k + 41 are
Euler's great interest in number theory can be traced to the influence of his friend in the St. Peterburg Academy, Christian Goldbach. A lot of his early work on number theory was based on the works of Pierre de Fermat, and developed some of Fermat's ideas. One focus of Euler's work was to link the nature of prime distribution with ideas in ...
In mathematics, a Heegner point is a point on a modular curve that is the image of a quadratic imaginary point of the upper half-plane. They were defined by Bryan Birch and named after Kurt Heegner , who used similar ideas to prove Gauss's conjecture on imaginary quadratic fields of class number one.
The term "Heegner numbers" is used by Conway and Guy in The ... , 40, is related to the Heegner number 163 = 4 · 41 − 1. Euler's formula, with taking ...