M. Karam Ouharou

The amplitudes program in quantum field theory has revealed new and unexpected connections between gauge theories and gravity. Through the double copy relationship, in which gravity amplitudes are closely tied to the squares of gauge theory amplitudes, it has become possible to compute gravity amplitudes which would otherwise be prohibitively complicated. A natural question is whether a similar relationship holds between exact solutions, in which solutions to general relativity can be generated from solutions to a gauge theory. Indeed, using the Kerr-Schild transformation, a simple and precise relationship can be drawn between gauge fields and spacetime metrics . It is striking that many exact solutions to the Einstein equations can be presented as the double copies of gauge theory solutions, including all four-dimensional black hole spacetimes. Generically, nonlinear behavior in general relativity makes a boundary-value approach very difficult . However, in electrodynamics this is the...

The representation of even numbers as the sum of two primes and the distribution of primes in short intervals were investigated and a main theorem was given out and proved, which states: For every number  n  greater than a positive number  n 0 , let  q  be an odd prime number smaller than  2 n − − √  and  d = 2 n − q , then there is always at least an odd number  d  which does not contain any prime factor smaller than  2 n − − √  and must be an odd prime number greater than  2 n − 2 n − − √ . Then it was proved that for every number  n  greater than 1, there are always at least a pair of primes  p  and  q  which are symmetrical about the number  n  so that even numbers greater than 2 can be expressed as the sum of two primes. Hence, the Goldbach's conjecture was proved. Also theorems of the distribution of primes in short intervals were given out and proved. By these theo...