On the Relationship Between Matiyasevich's and Smorynski's Theorems

Stiller proved that the Diophantine equation $x^2+119=15\cdot 2^{n}$ has exactly six solutions in positive integers. Motivated by this result we are interested in constructions of Diophantine equations of Ramanujan-Nagell type $x^2=Ak^{n}+B$ with many solutions. Here, $A,B\in\Z$ (thus $A, B$ are not necessarily positive) and $k\in\Z_{\geq 2}$ are given integers. In particular, we prove that for each $k$ there exists an infinite set $\cal{S}$ containing pairs of integers $(A, B)$ such that for each $(A,B)\in \cal{S}$ we have $\gcd(A,B)$ is square-free and the Diophantine equation $x^2=Ak^n+B$ has at least four solutions in positive integers. Moreover, we construct several Diophantine equations of the form $x^2=Ak^n+B$ with $k>2$, each containing five solutions in non-negative integers. %For example the equation $y^2=130\cdot 3^{n}+5550606$ has exactly five solutions with $n=0, 6, 11, 15, 16$. We also find new examples of equations $x^2=A2^{n}+B$ having six solutions in positive integers, e.g. the following Diophantine equations has exactly six solutions: \begin{equation*} \begin{array}{ll} x^2= 57\cdot 2^{n}+117440512 & n=0, 14, 16, 20, 24, 25, x^2= 165\cdot 2^{n}+26404 & n=0, 5, 7, 8, 10, 12. \end{array} \end{equation*} Moreover, based on an extensive numerical calculations we state several conjectures on the number of solutions of certain parametric families of the Diophantine equations of Ramanujan-Nagell type.

Mahler's measure is a natural generalization of Jensen's formula to polynomials in several variables. Its definition is as follows:The importance of Mahler's measure for polynomials in several variables lies in its connection to Lehmer's classical question which can be phrased in terms of Mahler's measure for polynomials in one variable:Given , are there any polynomials p with integer coefficients in one variable for which ?Surprisingly, Boyd [1] has shown that to answer this question, it is necessary to investigate the larger question involving polynomials in several variables.

Small solutions of linear Diophantine equations

In this paper, we study the Diophantine equation n x + 5 y = z 2 , where n is a positive integer and x, y, z are non-negative integers. We found that if n ≡ 1 (mod 4), then the Diophantine equation has no non-negative integer solution. If n ≡ 3 (mod 20) or n ≡ 7 (mod 20), then the Diophantine equation has all non-negative integer solutions, which are (n, x, y, z) = (n, 1, 0, (n+1)0.5), where (n+1)0.5 is a positive integer.

Let f(n)=1 if n=1, 2^(2^(n-2)) if n \in {2,3,4,5}, (2+2^(2^(n-4)))^(2^(n-4)) if n \in {6,7,8,...}. We conjecture that if a system T \subseteq {x_i+1=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} has only finitely many solutions in positive integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies x_1,...,x_n \leq f(n). We prove that the function f cannot be decreased and the conjecture implies that there is an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite. We show that if the conjecture is true, then this can be partially confirmed by the execution of a brute-force algorithm.

In this paper, we study the Diophantine equation (p+n)^x+p^y=z^2, where p, p+n are prime numbers and n is a positive integer such that n equiv mod 4. In case p=3 and n=4, Rao{7} showed that the non-negative integer solutions are (x,y,z)=(0,1,2) and (1,2,4) In case p>3 and pequiv 3pmod4, if n-1 is a prime number and 2n-1 is not prime number, then the non-negative integer solution (x, y, z) is (0, 1,\sqrt {p+1}) or ( 1, 0, \sqrt{p+n+1}). In case pequiv 1pmod4, the non-negative integer solution (x,y,z) is also (0, 1,\sqrt {p+1}) or ( 1,0, \sqrt{p+n+1}).

The definition of an invariant of finite order for links devised by Vassiliev [1] has led to the construction of a beautiful theory that includes almost all the known polynomial invariants. Bar-Natan [2] and Birman and Lin [3] proved that if two links have all their Vassiliev invariants equal, then their HOMFLY and Kauffman polynomials are also equal, and, as special cases of HOMFLY, so are the Jones and Conway polynomials and the Alexander polynomial in one variable. The Alexander polynomial in several variables (whose description in [4] by means of local relations is substantially more complicated) is missing from this list. The present note fills this gap. The Alexander polynomial in several variables is defined for ordered oriented links [5]. Each component has its own corresponding variable. For ordered links the definition of an invariant of finite order needs no modification. In the standard definition of the Alexander polynomial there is a lack of uniqueness in the choice of sign and in multiplication by a monomial of the form tj. This is related to the fact that the Alexander polynomial is an invariant of torsion type (see [6]) and creates difficulties in the study of its interrelations with invariants of finite order. We shall give below a definition of a normalized Alexander polynomial in several variables, which is uniquely defined by a diagram, and we shall show that after a certain change of variables all its coefficients are Vassiliev invariants. We shall give the definition straight away for links in R3 that are allowed to have finitely many points of self-intersection. We shall consider links with at least two components. We begin the definition of the normalized Alexander polynomial with a simple algebraic assertion. First let us introduce the following notation. For a matrix A = (dij)i^ij^n we denote by A = (2ij)i^ij Δ^(Λ), where Aji(A) is the minor complementary to a,;.

Number theory is the second large field of mathematics that comes to us from the Pythagoreans via Euclid. The Pythagorean theorem led mathematicians to the study of squares and sums of squares; Euclid drew attention to the primes by proving that there are infinitely many of them. Euclid’s investigations were based on the so-called Euclidean algorithm, a method for finding the greatest common divisor of two natural numbers. Common divisors are the key to basic results about prime numbers, in particular unique prime factorization, which says that each natural number factors into primes in exactly one way. Another discovery of the Pythagoreans, the irrationality of \(\sqrt{2}\), has repercussions in the world of natural numbers. Since\(\sqrt{2}\neq m/n\) for any natural numbers m, n, there is no solution of the equation \(x^2 - 2y^2 = 0\) in the natural numbers. But, surprisingly, there are natural number solutions of \(x^2 - \rm{2}y^2 = 1\), and in fact infinitely many of them. The same is true of the equation \(x^2 - Ny^2 = 1\) for any nonsquare natural number N. The latter equation, called Pell’s equation, is perhaps second in fame only to the Pythagorean equation \(x^2 + y^2 = z^2\), among equations for which integer solutions are sought. Methods for solving the Pell equation for general N were first discovered by Indian mathematicians, whose work we study in Chapter 5. Equations for which integer or rational solutions are sought are called Diophantine, after Diophantus. The methods he used to solve quadratic and cubic Diophantine equations are still of interest. We study his method for cubics in this chapter, and take it up again in Chapters 11 and 16.

The height of a rational number $p/q$ is denoted by $h(p/q)$ and equals $\text{max}(|p|,|q|)$ provided p/q is written in lowest terms. The height of a rational tuple $(x_1,...,x_n)$ is denoted by $h(x_1,...,x_n)$ and equals $\text{max}(h(x_1),...,h(x_n))$. Let $G_n={x_i+1=x_k: i,k \in {1,...,n}} \cup {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}$. Let $f(1)=1$, and let $f(n+1)=2^{(2^{(f(n))})}$ for every positive integer n. We conjecture: (1) if a system $S \subseteq G_n$ has only finitely many solutions in rationals $x_1,...,x_n$, then each such solution $(x_1,...,x_n)$ satisfies $h(x_1,...,x_n) \leq {1 (\text{if} n=1), 2^{(2^{(n-2)})} (\text{if} n>1)}$; (2) if a system $S \subseteq G_n$ has only finitely many solutions in non-negative rationals $x_1,...,x_n$, then each such solution $(x_1,...,x_n)$ satisfies $h(x_1,...,x_n) \leq f(2n)$. We prove: (1) both conjectures imply that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of rational solutions, if the solution set is finite; (2) both conjectures imply that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution.

Using a directed acyclic graph (dag) model of algorithms, we solve a problem related to precedence-constrained multiprocessor schedules for array computations: Given a sequence of dags and linear schedules parametrized by n, compute a lower bound on the number of processors required by the schedule as a function of n. In our formulation, the number of tasks that are scheduled for execution during any fixed time step is the number of non-negative integer solutions dn to a set of parametric linear Diophantine equations. We present an algorithm based on generating functions for constructing a formula for these numbers dn. The algorithm has been implemented as a Mathematica program. Example runs and the symbolic formulas for processor lower bounds automatically produced by the algorithm for Matrix-Vector Product, Triangular Matrix Product, and Gaussian Elimination problems are presented. Our approach actually solves the following more general problem: Given an arbitrary r× s integral matrix A and r-dimensional integral vectors b and c, let dn(n=0,1,…) be the number of solutions in non-negative integers to the system Az=nb+c. Calculate the (rational) generating function ∑n≥ 0dntn and construct a formula for dn.

Diophantine equations are gradually drawing attention in the study of hydrogen spectrum, eco- nomics, Biology, quantum Hall effect, chemistry, cryptography etc. Different types of schemes are employed to find solution of Diophantine equations. Some special types of Diophantine equations could be addressed with the help of Catalan’s conjecture and Congruence theory. The Diophantine equation (3x+63y=z2) is addressed in this paper to find the solution(s) in non-negative integers. It is found that the equation has only two solutions of (x,y,z) as (1,0,2) and (0,1,8) in non-negative integers.

Kronecker's theorem and Lehmer's problem for polynomials in several variables

In this paper, we consider the Diophantine equation (P+ 12) X+ (P+2K)=Z2where p > 3, p are primes and k is natural number, when x, y and z are non-negative integers. It is found that the Diophantine equation has no nonnegative integer solution.

In some recent investigations involving certain differential operators for a general family of Lagrange polynomials, Chan el al. encountered and proved a certain summation identity for the Lagrange polynomials in several variables. In the present paper, we derive some generalizations of this summation identity for the Chan-Chyan-Srivastava polynomials in several variables. We also discuss a number of interesting corollaries and consequences of our main results.

In this chapter we present the general properties of orthogonal polynomials in several variables, that is, those properties that hold for orthogonal polynomials associated with weight functions that satisfy some mild conditions but are not any more specific than that. This direction of study started with the classical work of Jackson [1936] on orthogonal polynomials in two variables. It was realized even then that the proper definition of orthogonality is in terms of polynomials of lower degree and that orthogonal bases are not unique. Most subsequent early work was focused on understanding the structure and theory in two variables. In Erdelyi et al . [1953], which documents the work up to 1950, one finds little reference to the general properties of orthogonal polynomials in more than two variables, other than (Vol. II, p. 265): “There does not seem to be an extensive general theory of orthogonal polynomials in several variables.” It was remarked there that the difficulty lies in the fact that there is no unique orthogonal system, owing to the many possible orderings of multiple sequences. And it was also pointed out that since a particular ordering usually destroys the symmetry, it is often preferable to construct biorthogonal systems. Krall and Sheffer [1967] studied and classified two-dimensional analogues of classical orthogonal polynomials as solutions of partial differential equations of the second order.