Families of symmetric periodic motions of the Euler problem

2005 ◽  
Vol 50 (4) ◽  
pp. 192-194
Author(s):  
V. N. Tkhai
Keyword(s):  
Periodic Motions ◽  
Euler Problem

PROCEEDURE FOR SOLVING GOLDBACH-EULER PROBLEM USING THE NUMBERS OF A SPECIAL TYPE

2018 ◽  
pp. 121-128
Author(s):  
Sergey Ivanovich Chermidov
Keyword(s):  
Prime Numbers ◽  
Euler Problem

Since the elements are closed, the set Θ = {6λ ± 1; λ ∈ N }, is a semigroup with respect to the operation of multiplication. The paper focuses on presenting even numbers ζ > 8 in the form of sums of two elements: θ1 = 6λ1 ± 1 and θ2 = 6λ2 ± 1 from the set Θ. Any even number ζ > 8 is comparable with one of the numbers m = (0, 2, -2), according to (mod 6). According to the remnants listed m , even numbers ζ > 8 are divided into 3 types. Each type has its own way of presenting sums in the form of two elements from the set Θ. For any even number ζ > 8 on the segment [1 ÷ ν] there is always at least a pair of numbers (λ1, λ2) ∈ [1 ÷ ν], that both are elements from the union of sets: the parameters of the prime numbers (twins) and the parameters (composite and prime) of numbers Θ. A variant of the solution of Goldbach - Euler conjecture for even numbers ζ > 8 is given on the set of primes P. Goldbach - Euler conjecture is also solvable in the set of prime numbers (twins), if the parameters of numbers θ1 and θ2, i.e. λ1 and λ2 belong to the set N \ FN , where FN is the set of parameters of the composite numbers Θ on the segment [1 ÷ ν].

scholarly journals Dynamical convexity of the Euler problem of two fixed centers

2017 ◽  
Vol 165 (2) ◽  
pp. 359-384 ◽  
Author(s):  
SEONGCHAN KIM
Keyword(s):  
Periodic Orbits ◽  
Universal Cover ◽  
Explicit Formulas ◽  
Energy Hypersurface ◽  
Collision Orbits ◽  
Euler Problem

AbstractWe give thorough analysis for the rotation functions of the critical orbits from which one can understand bifurcations of periodic orbits. Moreover, we give explicit formulas of the Conley–Zehnder indices of the interior and exterior collision orbits and show that the universal cover of the regularised energy hypersurface of the Euler problem is dynamically convex for energies below the critical Jacobi energy.

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