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A proof of the twin-prime conjecture, even in the stronger form of Hardy and Littlewood, namely that lim N →∞ 1 N p<N p,p+2 both prime log p · log(p + 2) = B 2 > 0, is presented using methods from classical analytic number theory.
The intensity of multiple echoes separated by a time 2 tau has been modeled using the closed form of a finite geometric series. This eliminates long exponential series, introduces the number of echoes as an independent parameter, and corrects for the net T1 relaxation during the echo train. Other sequences having echo trains can be modeled similarly.