The orbit intersection problem in positive characteristic

In this paper, we study the orbit intersection problem for the linear space and the algebraic group in positive characteristic. Let $K$ be an algebraically closed field of positive characteristic and let $\Phi_1, \Phi_{2}: K^d \longrightarrow K^{d}$ be affine maps, $\Phi_i({\bf x}) = A_i ({\bf x}) + {\bf x_i}$ (where each $A_i$ is a $d\times d$ matrix and ${\bf x}\in K^d$). If each ${\bf a}_i \in K^d$ is not $\Phi_i$-preperiodic, then we prove that the set $\left \{(n_1, n_2) \in \Z^{2} \mid \Phi_1^{n_1}({\bf a}_1) = \Phi_{2}^{n_{2}}({\bf a}_{2})\right\}$ is $p$-normal in $\mathbb{Z}^{2}$ of order at most $d$. Further, for the regular self-maps $\Phi_1, \Phi_{2}: \mathbb{G}_m^d \longrightarrow \mathbb{G}_m^d$, we show that the set $\{(n_1, n_{2}) \in \N_0^{2} \mid \Phi_1^{n_1}({\bf a}_1) = \Phi_{2}^{n_{2}}({\bf a}_{2})\}$ (where ${\bf a}_1, {\bf a}_2\in \mathbb{G}_m^d(K)$) lie in a finite number of linear and exponential one-parameter families. To do so, we use results on linear equations over multiplicative groups in positive characteristic and some results on systems of polynomial-exponential equations.Comment: 15 page

Some Generalizations and Properties of Balancing Numbers

The sequence of balancing numbers admits generalization in two different ways. The first way is through altering coefficients occurring in its binary recurrence sequence and the second way involves modification of its defining equation, thereby allowing more than one gap. The former generalization results in balancing-like numbers that enjoy all important properties of balancing numbers. The second generalization gives rise to gap balancing numbers and for each particular gap, these numbers are realized in multiple sequences. The definition of gap balancing numbers allow further generalization resulting in higher order gap balancing numbers but unlike gap balancing numbers, these numbers are scarce, the existence of these numbers are often doubtful. The balancing zeta function—a variant of Riemann zeta function—permits analytic continuation to the entire complex plane, while the series converges to irrational numbers at odd negative integers. The periods of balancing numbers modulo positive integers exhibits many wonderful properties. It coincides with the modulus of congruence if calculated modulo any power of two. There are three known primes such that the period modulo any one of these primes is equal to the period modulo its square. The sequence of balancing numbers remains stable modulo half of the primes, while modulo other half, the sequence is unstable

On perfect powers that are sum of two balancing numbers

Let $B_k$ denote the $k^{th}$ term of balancing sequence. In this paper we find all positive integer solutions of the Diophantine equation $B_n+B_m = x^q$ in variables $(m, n,x,q)$ under the assumption $n\equiv m \pmod 2$. Furthermore, we study the Diophantine equation $$B_n^{3}\pm B_m^{3} = x^q$$ with positive integer $q\geq 3$ and $\gcd(B_n, B_m) =1$.Comment: 9 page

Perfect powers in an alternating sum of consecutive cubes

In this paper, we consider the problem about finding out perfect powers in an alternating sum of consecutive cubes. More precisely, we completely solve the Diophantine equation (x+1)3 - (x+2)3 + ∙∙∙ - (x + 2d)3 + (x + 2d + 1)3 = zp, where p is prime and x,d,z are integers with 1 ≤ d ≤ 50

The Mordell-Weil bases for the elliptic curve y2=x3−m2x+m2y^2=x^3-m^2x+m^2

summary:Let $D_m$ be an elliptic curve over $\mathbb {Q}$ of the form $y^2 = x^3 -m^2x +m^2$, where $m$ is an integer. In this paper we prove that the two points $P_{-1}=(-m, m)$ and $P_0 = (0, m)$ on $D_m$ can be extended to a basis for $D_m(\mathbb {Q})$ under certain conditions described explicitly

Sums of S-units in recurrence sequences

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