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Let be the wonderful compactification of a connected adjoint semisimple group defined over a number field . We prove Manin’s conjecture on the asymptotic (as ) of the number of -rational points of of height less than , and give an explicit construction of a measure on , generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points on . Our approach is based on the mixing property of which we obtain with a rate of convergence.
The Manin conjecture is established for a split singular del Pezzo surface of degree four, with singularity type .
We prove Manin’s conjecture for a del Pezzo surface of degree six which has one singularity of type . Moreover, we achieve a meromorphic continuation and explicit expression of the associated height zeta function.
The Markoff conjecture states that given a positive integer , there is at most one triple of positive integers with that satisfies the equation . The conjecture is known to be true when is a prime power or two times a prime power. We present an elementary proof of this result. We also show that if in the class group of forms of discriminant , every ambiguous form in the principal genus corresponds to a divisor of , then the conjecture is true. As a result, we obtain criteria in terms of...
Let be the Lucas sequence. We show that the Diophantine equation has only the nonnegative integer solutions , , , , , , where is the th Mersenne number and .
In this paper, the author shows a technique of generating an infinite number of coprime integral solutions for of the Diophantine equation for any positive integral values of when (mod 6) or (mod 6). For doing this, we will be using a published result of this author in The Mathematics Student, a periodical of the Indian Mathematical Society.
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