The present paper proved that if λ1, λ2, λ3 are positive real numbers, λ1/λ2 is irrational. Then, the integer parts of λ1x12＋ λ2x22＋ λ3x34 are prime infinitely often for natural numbers x1, x2, x3.
We show that if λ1 , λ2 , λ3 are non-zero real numbers, not all of the same sign, η is real and λ1 /λ2 is irrational, then there are infinitely many ordered triples of primes (p1 , p2 , p3 ) for which |λ1 p1 + ...We show that if λ1 , λ2 , λ3 are non-zero real numbers, not all of the same sign, η is real and λ1 /λ2 is irrational, then there are infinitely many ordered triples of primes (p1 , p2 , p3 ) for which |λ1 p1 + λ2 p2 + λ3 p2 3 + η| < (max pj )- 1/40 (log max pj ) 4 .展开更多
Let Q be an infinite set of positive integers, τ 〉 1 be a real number and let Wτ（Q）={x∈R：｜x-p/q｜〈^-τ for infinitely many （p,q）∈ Z×Q}.For any given positive integer m, set Q（m）={n∈N：（n,m）=1}. ...Let Q be an infinite set of positive integers, τ 〉 1 be a real number and let Wτ（Q）={x∈R：｜x-p/q｜〈^-τ for infinitely many （p,q）∈ Z×Q}.For any given positive integer m, set Q（m）={n∈N：（n,m）=1}. If m is divisible by at least two prime factors, Adiceam [1] showed that Wτ（N） ／ Wτ（Q（m）） contains uncountably many Liouville numbers, and asked if it contains any non-Liouville numbers？ In this note, we give an affirmative answer to Adiceam＇s question.展开更多
In this article it is proved that there exist a large number of polynomials which have small discriminant in terms of the Euclidean and p-adic metrics simultaneously. The measure of the set of points which satisfy cer...In this article it is proved that there exist a large number of polynomials which have small discriminant in terms of the Euclidean and p-adic metrics simultaneously. The measure of the set of points which satisfy certain polynomial and derivative conditions is also determined.展开更多
In this paper we give several existence and effective results for theorems of Dirichletand Minkowski on simultaneous Diophantine approximation in the homogeneous case (includingp-adic and p-adic-real mixed cases).
This paper corresponds to the written versions of many lectures at several locations including the most recent one at Weinan Teachers University on June 8,2011.I would like to thank Professor Hailong Li for inviting m...This paper corresponds to the written versions of many lectures at several locations including the most recent one at Weinan Teachers University on June 8,2011.I would like to thank Professor Hailong Li for inviting me to publish this in the journal of his university.I wish also to express my deep gratitude to my friend Shigeru Kanemitsu,thanks to whom I could visit Weinan Teachers University,and who also came up with a written version of these notes. The topic is centered around the equation x2-dy2=±1,which is important because it produces the(infinitely many) units of real quadratic fields.This equation,where the unknowns x and y are positive integers while d is a fixed positive integer which is not a square,has been mistakenly called with the name of Pell by Euler.It was investigated by Indian mathematicians since Brahmagupta(628) who solved the case d=92,then by Bhaskara II(1150) for d=61 and Narayana(during the 14-th Century) for d=103.The smallest solution of x2-dy2=1 for these values of d are respectively 1 1512-92·1202=1, 1 766 319 0492-61·226 153 9802=1 and 227 5282-103·22 4192=1, and hence they could not have been found by a brute force search! After a short introduction to this long story of Pell's equation,we explain its connection with Diophantine approximation and continued fractions(which have close connection with the structure of real quadratic fields),and we conclude by saying a few words on more recent developments of the subject in terms of varieties.Finally we mention applications of continued fraction expansion to electrical circuits.展开更多
文摘The present paper proved that if λ1, λ2, λ3 are positive real numbers, λ1/λ2 is irrational. Then, the integer parts of λ1x12＋ λ2x22＋ λ3x34 are prime infinitely often for natural numbers x1, x2, x3.
基金Supported by the NNSF of China(11071070)Supported by the Science Research Plan of Education Department of Henan Province(2011B110002)
文摘We show that if λ1 , λ2 , λ3 are non-zero real numbers, not all of the same sign, η is real and λ1 /λ2 is irrational, then there are infinitely many ordered triples of primes (p1 , p2 , p3 ) for which |λ1 p1 + λ2 p2 + λ3 p2 3 + η| < (max pj )- 1/40 (log max pj ) 4 .
文摘Let Q be an infinite set of positive integers, τ 〉 1 be a real number and let Wτ（Q）={x∈R：｜x-p/q｜〈^-τ for infinitely many （p,q）∈ Z×Q}.For any given positive integer m, set Q（m）={n∈N：（n,m）=1}. If m is divisible by at least two prime factors, Adiceam [1] showed that Wτ（N） ／ Wτ（Q（m）） contains uncountably many Liouville numbers, and asked if it contains any non-Liouville numbers？ In this note, we give an affirmative answer to Adiceam＇s question.
基金supported by the Science Foundation Ireland Programme (Grant No. RFP/MTH1512)
文摘In this article it is proved that there exist a large number of polynomials which have small discriminant in terms of the Euclidean and p-adic metrics simultaneously. The measure of the set of points which satisfy certain polynomial and derivative conditions is also determined.
文摘In this paper we give several existence and effective results for theorems of Dirichletand Minkowski on simultaneous Diophantine approximation in the homogeneous case (includingp-adic and p-adic-real mixed cases).
文摘This paper corresponds to the written versions of many lectures at several locations including the most recent one at Weinan Teachers University on June 8,2011.I would like to thank Professor Hailong Li for inviting me to publish this in the journal of his university.I wish also to express my deep gratitude to my friend Shigeru Kanemitsu,thanks to whom I could visit Weinan Teachers University,and who also came up with a written version of these notes. The topic is centered around the equation x2-dy2=±1,which is important because it produces the(infinitely many) units of real quadratic fields.This equation,where the unknowns x and y are positive integers while d is a fixed positive integer which is not a square,has been mistakenly called with the name of Pell by Euler.It was investigated by Indian mathematicians since Brahmagupta(628) who solved the case d=92,then by Bhaskara II(1150) for d=61 and Narayana(during the 14-th Century) for d=103.The smallest solution of x2-dy2=1 for these values of d are respectively 1 1512-92·1202=1, 1 766 319 0492-61·226 153 9802=1 and 227 5282-103·22 4192=1, and hence they could not have been found by a brute force search! After a short introduction to this long story of Pell's equation,we explain its connection with Diophantine approximation and continued fractions(which have close connection with the structure of real quadratic fields),and we conclude by saying a few words on more recent developments of the subject in terms of varieties.Finally we mention applications of continued fraction expansion to electrical circuits.
基金The work of first author was partially supported by Natural Science Foundation of China second author's was partially supported by a UGC Grant of Hong Kong: Project No. 604103
文摘In this paper, we will introduce some problems and results between Diophantine approximation and value distribution theory.
文摘We discuss the analogue of the Nevanlinna theory and the theory of Diophantine approximation, focussing on the second main theorem and abc-conjecture.
