15 research outputs found
All functions g:N-->N which have a single-fold Diophantine representation are dominated by a limit-computable function f:N\{0}-->N which is implemented in MuPAD and whose computability is an open problem
Let E_n={x_k=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. For any
integer n \geq 2214, we define a system T \subseteq E_n which has a unique
integer solution (a_1,...,a_n). We prove that the numbers a_1,...,a_n are
positive and max(a_1,...,a_n)>2^(2^n). For a positive integer n, let f(n)
denote the smallest non-negative integer b such that for each system S
\subseteq E_n with a unique solution in non-negative integers x_1,...,x_n, this
solution belongs to [0,b]^n. We prove that if a function g:N-->N has a
single-fold Diophantine representation, then f dominates g. We present a MuPAD
code which takes as input a positive integer n, performs an infinite loop,
returns a non-negative integer on each iteration, and returns f(n) on each
sufficiently high iteration.Comment: 17 pages, Theorem 3 added. arXiv admin note: substantial text overlap
with arXiv:1309.2605. text overlap with arXiv:1404.5975, arXiv:1310.536
A hypothetical upper bound on the heights of the solutions of a Diophantine equation with a finite number of solutions
Let f(1)=1, and let f(n+1)=2^{2^{f(n)}} for every positive integer n. We
conjecture that if a system S \subseteq {x_i \cdot x_j=x_k: i,j,k \in
{1,...,n}} \cup {x_i+1=x_k: i,k \in {1,...,n}} has only finitely many solutions
in non-negative integers x_1,...,x_n, then each such solution (x_1,...,x_n)
satisfies x_1,...,x_n \leq f(2n). We prove: (1) the conjecture implies that
there exists 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, (2) the conjecture implies that the question whether or not a
Diophantine equation has only finitely many rational solutions is decidable
with an oracle for deciding whether or not a Diophantine equation has a
rational solution, (3) the conjecture implies that the question whether or not
a Diophantine equation has only finitely many integer solutions is decidable
with an oracle for deciding whether or not a Diophantine equation has an
integer solution, (4) the conjecture implies that if a set M \subseteq N has a
finite-fold Diophantine representation, then M is computable.Comment: 13 pages, section 7 expande
Is there a computable upper bound for the height of a solution of a Diophantine equation with a unique solution in positive integers?
Let B_n={x_i \cdot x_j=x_k, x_i+1=x_k: i,j,k \in {1,...,n}}. For a positive
integer n, let \xi(n) denote the smallest positive integer b such that for each
system S \subseteq B_n with a unique solution in positive integers x_1,...,x_n,
this solution belongs to [1,b]^n. Let g(1)=1, and let g(n+1)=2^{2^{g(n)}} for
every positive integer n. We conjecture that \xi(n) \leq g(2n) for every
positive integer n. We prove: (1) the function \xi: N\{0}-->N\{0} is computable
in the limit; (2) if a function f:N\{0}-->N\{0} has a single-fold Diophantine
representation, then there exists a positive integer m such that f(n)<\xi(n)
for every integer n>m; (3) the conjecture implies that there exists an
algorithm which takes as input a Diophantine equation D(x_1,...,x_p)=0 and
returns a positive integer d with the following property: for every positive
integers a_1,...,a_p, if the tuple (a_1,...,a_p) solely solves the equation
D(x_1,...,x_p)=0 in positive integers, then a_1,...,a_p \leq d; (4) the
conjecture implies that if a set M \subseteq N has a single-fold Diophantine
representation, then M is computable; (5) for every integer n>9, the inequality
\xi(n)<(2^{2^{n-5}}-1)^{2^{n-5}}+1 implies that 2^{2^{n-5}}+1 is composite.Comment: 10 pages. arXiv admin note: substantial text overlap with
arXiv:1309.268
Explicit Methods in Number Theory
These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics includes asymptotics for field extensions and class numbers, random matrices and L-functions, rational points on curves and higher-dimensional varieties, and aspects of lattice basis reduction
Artin's primitive root conjecture -a survey -
This is an expanded version of a write-up of a talk given in the fall of 2000
in Oberwolfach. A large part of it is intended to be understandable by
non-number theorists with a mathematical background. The talk covered some of
the history, results and ideas connected with Artin's celebrated primitive root
conjecture dating from 1927. In the update several new results established
after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer