Let $X$ be the wonderful compactification of a connected adjoint semisimple group $G$ defined over a number field $K$. We prove Manin’s conjecture on the asymptotic (as $T\to \infty$) of the number of $K$-rational points of $X$ of height less than $T$, and give an explicit construction of a measure on $X\left(𝔸\right)$, generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points $𝐆\left(K\right)$ on $X\left(𝔸\right)$. Our approach is based on the mixing property of $L^2(𝐆\left(K\right)\setminus 𝐆\left(𝔸\right))$ 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 ${\mathbf{A}}_4$.

We prove Manin’s conjecture for a del Pezzo surface of degree six which has one singularity of type ${\mathbf{A}}_2$. Moreover, we achieve a meromorphic continuation and explicit expression of the associated height zeta function.

The Markoff conjecture states that given a positive integer $c$, there is at most one triple $(a,b,c)$ of positive integers with $a\le b\le c$ that satisfies the equation $a^2+b^2+c^2=3abc$. The conjecture is known to be true when $c$ 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 $d=9c^2-4$, every ambiguous form in the principal genus corresponds to a divisor of $3c-2$, then the conjecture is true. As a result, we obtain criteria in terms of...

Let ${\left(L_n\right)}_{n\ge 0}$ be the Lucas sequence. We show that the Diophantine equation $L_n-L_m=M_k$ has only the nonnegative integer solutions $(n,m,k)=(2,0,1)$, $(3,1,2)$, $(3,2,1)$, $(4,3,2)$, $(5,3,3)$, $(6,2,4)$, $(6,5,3)$ where $M_k=2^k-1$ is the $k$th Mersenne number and $n>m$.

In this paper, the author shows a technique of generating an infinite number of coprime integral solutions for $(A,B,C)$ of the Diophantine equation $-(2^p·A^6)+B^3=C^2$ for any positive integral values of $p$ when $p\equiv 1$ (mod 6) or $p\equiv 2$ (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.