On the Positive Pell Equation

The binary quadratic equation () representing the hyperbola is studied for its non-zero distinct integer solutions. A few interesting properties among the solutions are presented. Employing the integer solutions of the equation under consideration, integer solutions for special straight lines, hyperbolas and parabolas are exhibited.

The convexity of the function y = E(x) defined by xy = yx

The set of solutions to the equation xy = yx has been studied extensively over the past three centuries, including work by well known mathematicians such as Daniel Bernoulli (1700–1782), Leonhard Euler (1707–1783), and Christian Goldbach (1690–1764). Various mathematicians have focused on the integer, rational, real, and complex solutions. For example, it has been shown (see [1]) that the equality 24 = 42 gives the only distinct integer solutions. Our exposition below presents some of the key ideas behind the positive real solutions to this equation and illustrates how rational solutions can be found. To learn more about the various solutions, the reader can consult the articles listed at the end of this paper, as well as the extensive references given in these articles. It is also possible to find some of this material on the Web.

ON THE TERNARY QUADRATIC DIOPHANTINE EQUATION 6z2 = 6x2 – 5y2

The ternary homogeneous quadratic equation given by 6z2 = 6x2 -5y2 representing a cone is analyzed for its non-zero distinct integer solutions. A few interesting relations between the solutions and special polygonal and pyramided numbers are presented. Also, given a solution, formulas for generating a sequence of solutions based on the given solutions are presented.

CONSTRUCTION OF IRRATIONAL GAUSSIAN DIOPHANTINE QUADRUPLES

Given any two non-zero distinct irrational Gaussian integers such that their product added with either 1 or 4 is a perfect square, an irrational Gaussian Diophantine quadruple ( , ) a0 a1, a2, a3 such that the product of any two members of the set added with either 1 or 4 is a perfect square by employing the non-zero distinct integer solutions of the system of double Diophantine equations. The repetition of the above process leads to the generation of sequences of irrational Gaussian Diophantine quadruples with the given property.

Statistical Distribution of Roots of a Polynomial Modulo Primes III

Let $f(x)=x^n+a_{n-1}x^{n-1}+\dots+a_0$ $(a_{n-1},\dots,a_0\in\mathbb Z)$ be a polynomial with complex roots $\alpha_1,\dots,\alpha_n$ and suppose that a linear relation over $\mathbb Q$ among $1,\alpha_1,\dots,\alpha_n$ is a multiple of $\sum_i\alpha_i+a_{n-1}=0$ only. For a prime number $p$ such that $f(x)\bmod p$ has $n$ distinct integer roots $0<r_1<\dots<r_n<p$, we proposed in a previous paper a conjecture that the sequence of points $(r_1/p,\dots,r_n/p)$ is equi-distributed in some sense. In this paper, we show that it implies the equi-distribution of the sequence of $r_1/p,\dots,r_n/p$ in the ordinary sense and give the expected density of primes satisfying $r_i/p<a$ for a fixed suffix $i$ and $0<a<1$.

ON THE POSITIVE PELL EQUATION y^2 = 72x^2 + 36

The binary quadratic equation represented by the positive pellian Y2 = 72X2 + 36 is analysed for its distinct integer solutions. A few interesting relations among the solutions are given. Further, employing the solutions of the above hyperbola, we have obtained solutions of other choices of hyperbolas, parabolas and special Pythagorean triangle.

On the Negative Pell Equation y2=45x2-11

The binary quadratic equation represented by the negative pellian y2=45x2-11 is analyzed for its distinct integer solutions. A few interesting relations among the solutions are also given. Further, employing the solutions of the above hyperbola, we have obtained solutions of other choices of hyperbolas, parabolas and special Pythagorean triangle.

Münchhausen Matrices

"The Baron's omni-sequence", $B(n)$, first defined by Khovanova and Lewis (2011), is a sequence that gives for each $n$ the minimum number of weighings on balance scales that can verify the correct labeling of $n$ identically-looking coins with distinct integer weights between $1$ gram and $n$ grams.A trivial lower bound on $B(n)$ is $\log_3 n$, and it has been shown that $B(n)$ is $\text{O}(\log n)$.We introduce new theoretical tools for the study of this problem, and show that $B(n)$ is $\log_3 n + \text{O}(\log \log n)$, thus settling in the affirmative a conjecture by Khovanova and Lewis that the true growth rate of the sequence is very close to the natural lower bound.

The Hausdorff dimension of small divisors for lower-dimensional KAM-tori

Given a bounded domain Ω ⊂ ℝ m and a Lipschitz map Φ : Ω ⟼ ℝ n , we determine the Hausdorff dimension of sets of points ω ∈ Ω for which the inequality | k ·ω — l·Φ(ω)| < Ψ (| k | + | l |) has infinitely many distinct integer solutions ( k, l )∈ℤ m x ℤ n satisfying | l | ⩽ h , where h is a fixed integer. These sets ‘interpolate’ between the cases h = 0 and h = ∞,which occur in the metric theory of Diophantine approximation of independent and dependent quantities, respectively. They arise, for example, in the perturbation theories of lower-dimensional tori in nearly integrable hamiltonian systems (KAM-theory). Among others, it turns out that their Hausdorff dimension is independent of h and n , it only depends on m and the lower order of Ψ at infinity. Part of this result even extends to the case n = ∞ of infinite co-dimension, which is relevant in the KAM-theory of certain nonlinear partial differential equations.