Diophantine equations in separated variables and lacunary polynomials
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DOI: 10.1142/s179304211750110x
Abstract: We study Diophantine equations of type f(x) = g(y), where f and g are lacunary polynomials. According to a well-known finiteness criterion, for a number field K and nonconstant f,g ∈ K[x], the equation f(x)… read more here.
Keywords: variables lacunary; equations separated; lacunary polynomials; separated variables ... See more keywords
Diophantine equations in primes: Density of prime points on affine hypersurfaces
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DOI: 10.1215/00127094-2021-0023
Abstract: Let F ∈ Z[x1, . . . , xn] be a homogeneous form of degree d ≥ 2, and let V ∗ F denote the singular locus of the affine variety V (F ) =… read more here.
Keywords: prime points; equations primes; density prime; points affine ... See more keywords
On the finiteness of solutions for polynomial-factorial Diophantine equations
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DOI: 10.1515/forum-2020-0138
Abstract: Abstract We study the Diophantine equations obtained by equating a polynomial and the factorial function, and prove the finiteness of integer solutions under certain conditions. For example, we show that there exist only finitely many… read more here.
Keywords: factorial diophantine; finiteness solutions; polynomial factorial; solutions polynomial ... See more keywords
On Diophantine equations involving Lucas sequences
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DOI: 10.1515/math-2019-0073
Abstract: Abstract In this paper, we shall study the Diophantine equation un = R(m)P(m)Q(m), where un is a Lucas sequence and R, P and Q are polynomials (under weak assumptions). read more here.
Keywords: involving lucas; lucas sequences; lucas; diophantine equations ... See more keywords
On the Diophantine equations x2 + 2α 3β 19γ = yn and x2 + 2α 3β 13γ = yn
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DOI: 10.1515/ms-2017-0243
Abstract: Abstract In this note we find all the solutions to the equation x2 + 2α 3β 19γ = yn in nonnegative unknowns with n ≥ 3 and gcd(x, y) = 1, and nonnegative solutions to… read more here.
Keywords: diophantine equations;
Solutions of Some Diophantine Equations in terms of Generalized Fibonacci and Lucas Numbers
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DOI: 10.15672/hjms.2017.527
Abstract: In this study, we present some identities involving generalized Fibonacci sequence $\left(U_{n}\right)$ and generalized Lucas sequence $\left(V_{n}\right)$. Then we give all solutions of the Diophantine equations $x^{2}-V_{n}xy+(-1)^{n}y^{2}=\pm (p^{2}+4)U_{n}^{2},$ $x^{2}-V_{n}xy+(-1)^{n}y^{2}=\pm U_{n}^{2},$ $x^{2}-(p^{2}+4)U_{n}xy-(p^{2}+4)(-1)^{n}y^{2}=\pm V_{n}^{2},$ $x^{2}-V_{n}xy\pm y^{2}=\pm 1,$… read more here.
Keywords: left right; equations terms; generalized fibonacci; solutions diophantine ... See more keywords
Teaching Congruences in Connection with Diophantine Equations
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DOI: 10.3390/educsci11090538
Abstract: The presented paper is devoted to the new teaching model of congruences of computer science students within the subject of discrete mathematics at universities. The main goal was to create a new model of teaching… read more here.
Keywords: diophantine equations; connection diophantine; teaching congruences; mathematics ... See more keywords
On Diophantine Equations Related to Order of Appearance in Fibonacci Sequence
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DOI: 10.3390/math7111073
Abstract: Let F n be the nth Fibonacci number. Order of appearance z ( n ) of a natural number n is defined as smallest natural number k, such that n divides F k . In… read more here.
Keywords: fibonacci; order; equations related; order appearance ... See more keywords
On Diophantine Equations Related to Narayana's Cows Sequence and Double Factorials or Repdigits
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DOI: 10.3390/sym14102004
Abstract: In this paper, we determine all the Narayana’s cows numbers that are factorials or double factorials. We also show that 88 is the only repdigit (i.e., a class of numbers that has reflectional symmetry across… read more here.
Keywords: related narayana; cows sequence; diophantine equations; narayana cows ... See more keywords