7 research outputs found
Computational sieving applied to some classical number-theoretic problems
Many problems in computational number theory require the application of some sieve. Efficient implementation of these sieves on modern computers has extended our knowledge of these problems considerably. This is illustrated by three classical problems: the Goldbach conjecture, factoring large numbers, and computing the summatory function of the M'{obius function
A primordial, mathematical, logical and computable, demonstration (proof) of the family of conjectures known as GoldbachÂŽs
licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.In
this
document,
by
means
of
a
novel
system
model
and
first
order
topological,
algebraic
and
geometrical
free-Ââcontext
formal
language
(NT-ÂâFS&L),
first,
we
describe
a
new
signature
for
a
set
of
the
natural
numbers
that
is
rooted
in
an
intensional
inductive
de-Ââembedding
process
of
both,
the
tensorial
identities
of
the
known
as
ânatural
numbersâ,
and
the
abstract
framework
of
theirs
locus-Ââpositional
based
symbolic
representations.
Additionally,
we
describe
that
NT-ÂâFS&L
is
able
to:
i.-Ââ
Embed
the
De
MorganÂŽs
Laws
and
the
FOL-ÂâPeanoÂŽs
Arithmetic
Axiomatic.
ii.-Ââ
Provide
new
points
of
view
and
perspectives
about
the
succession,
precede
and
addition
operations
and
of
their
abstract,
topological,
algebraic,
analytic
geometrical,
computational
and
cognitive,
formal
representations.
Second,
by
means
of
the
inductive
apparatus
of
NT-ÂâFS&L,
we
proof
that
the
family
of
conjectures
known
as
Glodbachâs
holds
entailment
and
truth
when
the
reasoning
starts
from
the
consistent
and
finitary
axiomatic
system
herein
describedWe
wish
to
thank
the
Organic
Chemistry
Institute
of
the
Spanish
National
Research
Council
(IQOG/CSIC)
for
its
operative
and
technical
support
to
the
Pedro
Noheda
Research
Group
(PNRG).
We
also
thank
the
Institute
for
Physical
and
Information
Technologies
(ITETI/CSIC)
of
the
Spanish
National
Research
Council
for
their
hospitality.
We
also
thank
for
their
long
years
of
dedicated
and
kind
support
Dr.
Juan
MartĂnez
Armesto
(VATC/CSIC),
Belén
Cabrero
SuĂĄrez
(IQOG/CSIC,
Administration),
Mar
Caso
Neira
(IQOG/CENQUIOR/CSIC,
Library)
and
David
Herrero
RuĂz
(PNRG/IQOG/CSIC).
We
wish
to
thank
to
BernabĂ©-ÂâPajaresÂŽs
brothers
(Dr.
Manuel
BernabĂ©-ÂâPajares,
IQOG/CSIC
Structural
Chemistry
&
Biochemistry;
Magnetic
Nuclear
Resonance
and
Dr.
Alberto
Bernabé
Pajares
(Greek
Philology
and
Indo-ÂâEuropean
Linguistics/UCM),
for
their
kind
attention
during
numerous
and
kind
discussions
about
space,
time,
imaging
and
representation
of
knowledge,
language,
transcription
mistakes,
myths
and
humans
always
holding
us
familiar
illusion
and
passion
for
knowledge
and
intellectual
progress.
We
wish
to
thank
Dr.
Carlos
Cativiela
MarĂn
(ISQCH/UNIZAR)
for
his
encouragement
and
for
kind
listening
and
attention.
We
wish
to
thank
Miguel
Lorca
Melton
for
his
encouragement
and
professional
point
of
view
as
Patent
Attorney.
Last
but
not
least,
our
gratitude
to
Nati,
MarĂa
and
Jaime
for
the
time
borrowed
from
a
loving
husband
and
father.
Finally,
we
apologize
to
many
who
have
not
been
mentioned
today,
but
to
whom
we
are
grateful.
Finally,
let
us
point
out
that
we
specially
apologize
to
many
who
have
been
mentioned
herein
for
any
possible
misunderstanding
regarding
the
sense
and
intension
of
their
philosophic,
scientific
and/or
technical
hard
work
and
milestone
ideas;
we
hope
that
at
least
Goldbach,
Euler
and
Feymann
do
not
belong
to
this
last
humanÂŽs
collectivity.Peer reviewe
New experimental results concerning the Goldbach conjecture
The Goldbach conjecture states that every even integer can be written as a sum of two prime numbers. It is known to be true up to 4times 10^{11. In this paper, new experiments on a Cray C916 supercomputer and on an SGI compute server with 18 R8000 CPUs are described, which extend this bound to 10^{14. Two consequences are that (1) under the assumption of the Generalized Riemann hypothesis, every odd number can be written as a sum of three prime numbers, and (2) under the assumption of the Riemann hypothesis, every even positive integer can be written as a sum of at most four prime numbers. In addition, we have verified the Goldbach conjecture for all the even numbers in the intervals [10^{5i, 10^{5i+10^8], for and [10^{10i, 10^{10i+10^9], for . A heuristic model is given which predicts the average number of steps needed to verify the Goldbach conjecture on a given interval. Our experimental results are in good agreement with this prediction. This adds to the evidence of the truth of the Goldbach conjecture