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1,602
result(s) for
"Diophantine equation"
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Solving 𝑆-unit, Mordell, Thue, Thue–Mahler and Generalized Ramanujan–Nagell Equations via the Shimura–Taniyama Conjecture
In the first part we construct algorithms (over
In the second part we establish new results for certain old Diophantine problems (e.g. the difference of
squares and cubes) related to Mordell equations, and we prove explicit height bounds for cubic Thue, cubic Thue–Mahler and generalized
Ramanujan–Nagell equations. As a byproduct, we obtain here an alternative proof of classical theorems of Baker, Coates and
Vinogradov–Sprindžuk. In fact we get refined versions of their theorems, which improve the actual best results in many fundamental
cases. We also conduct some effort to work out optimized height bounds for
In the third part we solve the problem of constructing an
efficient sieve for the
eBook
On prime powers in linear recurrence sequences
In this paper we consider the Diophantine equation
U
n
=
p
x
where
U
n
is a linear recurrence sequence,
p
is a prime number, and
x
is a positive integer. Under some technical hypotheses on
U
n
, we show that, for any
p
outside of an effectively computable finite set of prime numbers, there exists at most one solution (
n
,
x
) to that Diophantine equation. We compute this exceptional set for the Tribonacci sequence and for the Lucas sequence plus one.
Journal Article
A NOTE ON THE DIOPHANTINE EQUATION qx + py = z2
In this paper, we investigate the non-negative integer solutions of the equation of the form qx + py = z2 for x, y and z, with p, q primes. In particular, we consider the equation 3x + py = z2, with p a prime congruent to 5 modulo 12. We prove that (1, 0, 2) is the unique non-negative integer solution of this equation. Moreover, we prove that (1, 0, 2) is also the unique non-negative integer solution for the equation 3x + by = z2 where b is a positive integer congruent to 1 modulo 4 and has a prime factor congruent to 5 modulo 12 or congruent to 7 modulo 12.
Journal Article
THE PAUCITY PROBLEM FOR CERTAIN SYMMETRIC DIOPHANTINE EQUATIONS
Let
$\\varphi _1,\\ldots ,\\varphi _r\\in {\\mathbb Z}[z_1,\\ldots z_k]$
be integral linear combinations of elementary symmetric polynomials with
$\\text {deg}(\\varphi _j)=k_j\\ (1\\le j\\le r)$
, where
$1\\le k_1
Journal Article
Integer Solutions to Some Diophantine Equations of Leech Type with Geometric Applications
In this paper, we derive integer pseudo-parametric solutions to two sets of Diophantine equations. Moreover, we describe the so-called Double Crossed Ladder (DCL) and show how these results can be used to calculate an infinite number of integer solutions of its sides. In addition, we describe the fact that these results can be used to derive some corresponding sets of integer sides of more complex geometric structures.
Journal Article
On ternary Diophantine equations of signature $(p,p,\\text{3})$ over number fields
In this paper, we prove results about solutions of the Diophantine equation
$x^p+y^p=z^3$
over various number fields using the modular method. First, by assuming some standard modularity conjecture, we prove an asymptotic result for general number fields of narrow class number one satisfying some technical conditions. Second, we show that there is an explicit bound such that the equation
$x^p+y^p=z^3$
does not have a particular type of solution over
$K=\\mathbb {Q}(\\sqrt {-d})$
, where
$d=1,7,19,43,67$
whenever p is bigger than this bound. During the course of the proof, we prove various results about the irreducibility of Galois representations, image of inertia groups, and Bianchi newforms.
Journal Article
DIOPHANTINE EQUATIONS FOR POLYNOMIALS WITH RESTRICTED COEFFICIENTS, I (POWER VALUES)
We give effective finiteness results for the power values of polynomials with coefficients composed of a fixed finite set of primes; in particular, of Littlewood polynomials.
Journal Article
Tian’s Conjecture on the Prime Factorization of the Binomial Coefficient (n+12)
Tian’s conjecture states that for any fixed distinct prime numbers p1,…,pm, the Diophantine equation n+12=p1α1·p2α2···pmαm in positive integers n,α1,…,αm has at most m solutions. In this paper, we develop a computational method to verify some special cases of this conjecture. We also give an alternative proof using the classical Zsigmondy theorem. For m=2 and 3, a sharp absolute upper bound for the number of solutions is given.
Journal Article