Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for ...Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for anyε>0,the number of v∈V with v≤N such that the following inequality|λ_(1)p_(1)~2+λ_(2)p_(2)~2+λ_(3)p_(3)~4+λ_(4)p_(4)~4+λ_5p_5~4-v|<v^(-δ)has no solution in prime variables p_(1),p_(2),p_(3),p_(4),p_(5)does not exceed O(N^(29/32+2δ+ε)).展开更多
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
Let ξ be an irrational number with simple continued fraction expansion ξ = [a0;a1,··· ,ai,···] and pi be its ith convergent. Let Ci be de?ned by ξ ? pi = (?1)i/(Ciqiqi ). The qi qi +1 ...Let ξ be an irrational number with simple continued fraction expansion ξ = [a0;a1,··· ,ai,···] and pi be its ith convergent. Let Ci be de?ned by ξ ? pi = (?1)i/(Ciqiqi ). The qi qi +1 author proves the following theorem: Theorem. Let r > 1,R > 1 be two real numbers and q L = 1 + 1 + anan rR, K = 1 L + L2 ? 4 . r?1 R?1 +1 2 (r?1)(R?1) Then (i) Cn < r, Cn < R imply Cn > K; ?2 ?1 (ii) Cn > r, Cn > R imply Cn < K. ?2 ?1 This theorem generalizes the main result in [1].展开更多
In 2004,Tong found bounds for the approximation quality of a regular continued fraction convergent to a rational number,expressed in bounds for both the previous and next approximation.The authors sharpen his results ...In 2004,Tong found bounds for the approximation quality of a regular continued fraction convergent to a rational number,expressed in bounds for both the previous and next approximation.The authors sharpen his results with a geometric method and give both sharp upper and lower bounds.The asymptotic frequencies that these bounds occur are also calculated.展开更多
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
This paper proposes three new attacks. In the first attack we consider the class of the public exponents satisfying an equation e X-N Y ＋（ap^r＋ bq^r）Y = Z for suitably small positive integers a, b. Applying contin...This paper proposes three new attacks. In the first attack we consider the class of the public exponents satisfying an equation e X-N Y ＋（ap^r＋ bq^r）Y = Z for suitably small positive integers a, b. Applying continued fractions we show thatY/Xcan be recovered among the convergents of the continued fraction expansion of e/N. Moreover, we show that the number of such exponents is at least N^（2/（r＋1）-ε）where ε≥ 0 is arbitrarily small for large N. The second and third attacks works upon k RSA public keys（N_i, e_i） when there exist k relations of the form e_ix-N_iy_i ＋（ap_i^r ＋ bq_i^r ）y_i = z_i or of the form e_ix_i-N_iy ＋（ap_i^r ＋ bq_i^r ）y = z_i and the parameters x, x_i, y, y_i, z_i are suitably small in terms of the prime factors of the moduli. We apply the LLL algorithm, and show that our strategy enables us to simultaneously factor k prime power RSA moduli.展开更多
LetΨ={ψn}n≥1 be an iterated function system(IFS)on[0,1]with attractor J.Associated with each x∈J,there is a sequence{ωn(x)}n≥1 consisting of integers,called the digit sequence of x,such that■((1))We revisit the...LetΨ={ψn}n≥1 be an iterated function system(IFS)on[0,1]with attractor J.Associated with each x∈J,there is a sequence{ωn(x)}n≥1 consisting of integers,called the digit sequence of x,such that■((1))We revisit the Borel-Bernstein theorem in a d-decaying Gauss-like IFS,and completely characterize the metrical properties of the set■whereΦ:ℕ→ℝis a positive function.展开更多
In this paper, we establish a Second Main Theorem for an algebraically degenerate holomorphic curve f : C → Pn(C) intersecting hypersurfaces in general position. The related Diophantine problems are also considered.
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
基金Supported by NSFC(Nos.12301006,12471009,12071238,11901566,12001047,11971476)Beijing Natural Science Foundation(No.1242003)。
文摘Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for anyε>0,the number of v∈V with v≤N such that the following inequality|λ_(1)p_(1)~2+λ_(2)p_(2)~2+λ_(3)p_(3)~4+λ_(4)p_(4)~4+λ_5p_5~4-v|<v^(-δ)has no solution in prime variables p_(1),p_(2),p_(3),p_(4),p_(5)does not exceed O(N^(29/32+2δ+ε)).
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Let ξ be an irrational number with simple continued fraction expansion ξ = [a0;a1,··· ,ai,···] and pi be its ith convergent. Let Ci be de?ned by ξ ? pi = (?1)i/(Ciqiqi ). The qi qi +1 author proves the following theorem: Theorem. Let r > 1,R > 1 be two real numbers and q L = 1 + 1 + anan rR, K = 1 L + L2 ? 4 . r?1 R?1 +1 2 (r?1)(R?1) Then (i) Cn < r, Cn < R imply Cn > K; ?2 ?1 (ii) Cn > r, Cn > R imply Cn < K. ?2 ?1 This theorem generalizes the main result in [1].
文摘In 2004,Tong found bounds for the approximation quality of a regular continued fraction convergent to a rational number,expressed in bounds for both the previous and next approximation.The authors sharpen his results with a geometric method and give both sharp upper and lower bounds.The asymptotic frequencies that these bounds occur are also calculated.
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.
文摘This paper proposes three new attacks. In the first attack we consider the class of the public exponents satisfying an equation e X-N Y ＋（ap^r＋ bq^r）Y = Z for suitably small positive integers a, b. Applying continued fractions we show thatY/Xcan be recovered among the convergents of the continued fraction expansion of e/N. Moreover, we show that the number of such exponents is at least N^（2/（r＋1）-ε）where ε≥ 0 is arbitrarily small for large N. The second and third attacks works upon k RSA public keys（N_i, e_i） when there exist k relations of the form e_ix-N_iy_i ＋（ap_i^r ＋ bq_i^r ）y_i = z_i or of the form e_ix_i-N_iy ＋（ap_i^r ＋ bq_i^r ）y = z_i and the parameters x, x_i, y, y_i, z_i are suitably small in terms of the prime factors of the moduli. We apply the LLL algorithm, and show that our strategy enables us to simultaneously factor k prime power RSA moduli.
基金supported by the Scientific Research Project of Colleges and Universities in Anhui Province(2024AH050016)supported by the NSFC(12171172).The third author was supported by the NSFC(12201476)the Fundamental Research Funds for the Central Universities.
文摘LetΨ={ψn}n≥1 be an iterated function system(IFS)on[0,1]with attractor J.Associated with each x∈J,there is a sequence{ωn(x)}n≥1 consisting of integers,called the digit sequence of x,such that■((1))We revisit the Borel-Bernstein theorem in a d-decaying Gauss-like IFS,and completely characterize the metrical properties of the set■whereΦ:ℕ→ℝis a positive function.
基金supported by National Natural Science Foundation of China (Grant Nos. 11171255, 10901120)Doctoral Program Foundation of the Ministry of Education of China (Grant No.20090072110053)US National Security Agency (Grant Nos. H98230-09-1-0004, H98230-11-1-0201)
文摘In this paper, we establish a Second Main Theorem for an algebraically degenerate holomorphic curve f : C → Pn(C) intersecting hypersurfaces in general position. The related Diophantine problems are also considered.
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.
