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Pythagorician Divisors and Applications to Some Diophantine Equations
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作者 François Emmanuel Tanoé Prosper Kouadio Kimou 《Advances in Pure Mathematics》 2023年第2期35-70,共36页
We consider the Pythagoras equation X<sup>2</sup> +Y<sup>2</sup> = Z<sup>2</sup>, and for any solution of the type (a,b = 2<sup>s</sup>b<sub>1 </sub>≠0,c) ... We consider the Pythagoras equation X<sup>2</sup> +Y<sup>2</sup> = Z<sup>2</sup>, and for any solution of the type (a,b = 2<sup>s</sup>b<sub>1 </sub>≠0,c) ∈ N<sup>*3</sup>, s ≥ 2, b<sub>1</sub>odd, (a,b,c) ≡ (±1,0,1)(mod 4), c > a , c > b, and gcd(a,b,c) = 1, we then prove the Pythagorician divisors Theorem, which results in the following: , where (d,d′′) (resp. (e,e<sup>n</sup>)) are unique particular divisors of a and b, such that a = dd′′ (resp. b = ee′′ ), these divisors are called: Pythagorician divisors from a, (resp. from b). Let’s put λ ∈{0,1}, defined by: and S = s -λ (s -1). Then such that . Moreover the map is a bijection. We apply this new tool to obtain a new classification of the primitive, positive and non-trivial solutions of the Pythagoras equations: a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup> via the Pythagorician parameters (d,e,S ). We obtain for (d, e) fixed, the equivalence class of any Pythagorician solution (a,b,c), checking , namely: . We also update the solutions of some Diophantine equations of degree 2, already known, but very important for the resolution of other equations. With this tool of Pythagorean divisors, we have obtained (in another paper) new recurrent methods to solve Fermat’s equation: a<sup>4</sup> + b<sup>4 </sup>= c<sup>4</sup>, other than usual infinite descent method;and to solve congruent numbers problem. We believe that this tool can bring new arguments, for Diophantine resolution, of the general equations of Fermat: a<sup>2p</sup> + b<sup>2p</sup> = c<sup>2p</sup> and a<sup>p</sup> + b<sup>p</sup> = c<sup>p</sup>. MSC2020-Mathematical Sciences Classification System: 11A05-11A51-11D25-11D41-11D72. 展开更多
关键词 Pythagoras equation Pythagorician Triplets diophantine equations of Degree 2 Factorisation-Gcd-Fermat’s equations
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Diophantine Quotients and Remainders with Applications to Fermat and Pythagorean Equations
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作者 Prosper Kouadio Kimou François Emmanuel Tanoé 《American Journal of Computational Mathematics》 2023年第1期199-210,共12页
Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pyth... Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pythagoras’- Fermat’s equation defined as follows.                                                                                         (1) when , it is well known that this equation has an infinity of solutions but has none (non-trivial) when . We also know that the last result, named Fermat-Wiles theorem (or FLT) was obtained at great expense and its understanding remains out of reach even for a good fringe of professional mathematicians. The aim of this research is to set up new simple but effective tools in the treatment of Diophantine equations and that of Pythagoras-Fermat. The tools put forward in this research are the properties of the quotients and the Diophantine remainders which we define as follows. Let a non-trivial triplet () solution of Equation (1) such that . and are called the Diophantine quotients and remainders of solution . We compute the remainder and the quotient of b and c by a using the division algorithm. Hence, we have: and et with . We prove the following important results. if and only if and if and only if . Also, we deduce that or for any hypothetical solution . We illustrate these results by effectively computing the Diophantine quotients and remainders in the case of Pythagorean triplets using a Python program. In the end, we apply the previous properties to directly prove a partial result of FLT. . 展开更多
关键词 diophantine equation Modular Arithmetic Fermat-Wiles Theorem Pythagorean Triplets Division Theorem Division Algorithm Python Program diophantine Quotients diophantine Remainders
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A Note on Diophantine Equation Ax^4+ 1 = By^2 and Erds' Conjecture on Combinatorial Number
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作者 曹珍富 《Journal of Harbin Institute of Technology(New Series)》 EI CAS 1998年第2期1-3,共3页
A necessary and suffcient condition is given for the equation Ax4+ 1 =By2 to have positive integer solution, and an effective method is derived for solving equation a2x4 + 1 = By2 in positive integers x, y for given h... A necessary and suffcient condition is given for the equation Ax4+ 1 =By2 to have positive integer solution, and an effective method is derived for solving equation a2x4 + 1 = By2 in positive integers x, y for given ho and B completely. Also, using a recently result of Ribet, Darmon and Merel, we proved that Erdos’ conjecture on combinatorial number is right. 展开更多
关键词 diophantine equations SEQUENCE combinatorial number
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On Diophantine Equation X(X+1)(X+2)(X+3)=14Y(Y+1)(Y+2)(Y+3) 被引量:2
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作者 段辉明 郑继明 《Journal of Southwest Jiaotong University(English Edition)》 2009年第1期90-93,共4页
The Diophantine equation X( X + 1 ) ( X + 2 ) ( X + 3 ) = 14Y( Y + 1 ) ( Y + 2 ) ( Y + 3 ) still remains open. Using recurrence sequence, Maple software, Pell equation and quadraric residue, this pap... The Diophantine equation X( X + 1 ) ( X + 2 ) ( X + 3 ) = 14Y( Y + 1 ) ( Y + 2 ) ( Y + 3 ) still remains open. Using recurrence sequence, Maple software, Pell equation and quadraric residue, this paper proved it has only two positive integer solutions, i. e., (X,Y) = (5,2) ,(7,3). 展开更多
关键词 Integer solution diophantine equation Recurrent sequence Quadratic residue
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A Kind of Diophantine Equations in Finite Simple Groups 被引量:3
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作者 曹珍富 《Northeastern Mathematical Journal》 CSCD 2000年第4期391-397,共7页
In this paper, we prove that if p, q are distinct primes, (p,q)≡(1,7) (mod 12) and Legendres symbol pq=1 , then the equation 1+p a=2 bq c+2 dp eq f has only solutions of the form (a,b,c,d,e,f)=... In this paper, we prove that if p, q are distinct primes, (p,q)≡(1,7) (mod 12) and Legendres symbol pq=1 , then the equation 1+p a=2 bq c+2 dp eq f has only solutions of the form (a,b,c,d,e,f)=(t,0,0,0,t,0), where t is a non negative integer. We also give all solutions of a kind of generalized Ramanujan Nagell equations by using the theories of imaginary quadratic field and Pells equation. 展开更多
关键词 exponential diophantine equation generalized Ramanujan Nagell equation finite simple group difference set
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Complete solution of the diophantine equation A^2x^4-By^2=1 and some related problems 被引量:1
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作者 曹珍富 《Journal of Harbin Institute of Technology(New Series)》 EI CAS 2001年第2期108-110,共3页
We prove that diophantine equation in title has at most one positive integer solution for any positive integers A>1, B>1. It follows that Lucas problem is very simple to solve and a recent result of Bennett ... We prove that diophantine equation in title has at most one positive integer solution for any positive integers A>1, B>1. It follows that Lucas problem is very simple to solve and a recent result of Bennett is very simple to prove. 展开更多
关键词 quartic diophantine equation COMPUTER C++
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Heron Triangle and Diophantine Equation
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作者 YANG Shi-chun MA Chang- wei 《Chinese Quarterly Journal of Mathematics》 CSCD 北大核心 2005年第3期242-246,共5页
In this paper, we study the quantic Diophantine equation (1) with elementary geometry method, therefore all positive integer solutions of the equation (1) are obtained, and existence of Heron triangle whose median... In this paper, we study the quantic Diophantine equation (1) with elementary geometry method, therefore all positive integer solutions of the equation (1) are obtained, and existence of Heron triangle whose median lengths are all positive integer are discussed here. 展开更多
关键词 quantic diophantine equation positive integer solution Heron triangle MEDIAN
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A New Proof of Diophantine Equation ■
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作者 ZHU HUI-LIN 《Communications in Mathematical Research》 CSCD 2009年第3期282-288,共7页
By using algebraic number theory and p-adic analysis method, we give a new and simple proof of Diophantine equation (^n2) = (^m4)
关键词 binomial diophantine equation fundamental unit FACTORIZATION p-adic analysis method
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On the Diophantine Equation y^2= px(x^2+ 2)
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作者 WANG Xiao-ying 《Chinese Quarterly Journal of Mathematics》 CSCD 2009年第4期499-503,共5页
For any fixed odd prime p, let N(p) denote the number of positive integer solutions (x, y) of the equation y^2 = px(x^2 + 2). In this paper, using some properties of binary quartic Diophantine equations, we pro... For any fixed odd prime p, let N(p) denote the number of positive integer solutions (x, y) of the equation y^2 = px(x^2 + 2). In this paper, using some properties of binary quartic Diophantine equations, we prove that ifp ≡ 5 or 7(mod 8), then N(p) = 0; ifp ≡ 1(mod 8), then N(p) 〈 1; if p〉 3 andp ≡ 3(rood 8), then N(p) ≤ 2. 展开更多
关键词 cubic and quartic diophantine equation number of solutions upper bound 2000 MR Subject Classification: 11D25
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A Variant of Fermat’s Diophantine Equation
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作者 Serdar Beji 《Advances in Pure Mathematics》 2021年第12期929-936,共8页
A variant of Fermat’s last Diophantine equation is proposed by adjusting the number of terms in accord with the power of terms and a theorem describing the solubility conditions is stated. Numerically obtained primit... A variant of Fermat’s last Diophantine equation is proposed by adjusting the number of terms in accord with the power of terms and a theorem describing the solubility conditions is stated. Numerically obtained primitive solutions are presented for several cases with number of terms equal to or greater than powers. Further, geometric representations of solutions for the second and third power equations are devised by recasting the general equation in a form with rational solutions less than unity. Finally, it is suggested to consider negative and complex integers in seeking solutions to Diophantine forms in general. 展开更多
关键词 Variant of Fermat’s Last equation Positive Integer Solutions of New Fermat-Type equations Geometric Representations for Solutions of New diophantine equations
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Non-Negative Integer Solutions of Two Diophantine Equations 2x + 9y = z2 and 5x + 9y = z2
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作者 Md. Al-Amin Khan Abdur Rashid Md. Sharif Uddin 《Journal of Applied Mathematics and Physics》 2016年第4期762-765,共4页
In this paper, we study two Diophantine equations of the type p<sup>x</sup> + 9<sup>y</sup> = z<sup>2</sup> , where p is a prime number. We find that the equation 2<sup>x</... In this paper, we study two Diophantine equations of the type p<sup>x</sup> + 9<sup>y</sup> = z<sup>2</sup> , where p is a prime number. We find that the equation 2<sup>x</sup> + 9<sup>y</sup> = z<sup>2</sup> has exactly two solutions (x, y, z) in non-negative integer i.e., {(3, 0, 3),(4, 1, 5)} but 5<sup>x</sup> + 9<sup>y</sup> = z<sup>2</sup> has no non-negative integer solution. 展开更多
关键词 Exponential diophantine equation Integer Solutions
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Diophantine Equations and the Freeness of Mobius Groups
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作者 Marin Gutan 《Applied Mathematics》 2014年第10期1400-1411,共12页
Let p and q be two fixed non zero integers verifying the condition gcd(p,q) = 1. We check solutions in non zero integers a1,b1,a2,b2 and a3 for the following Diophantine equations: (B1) (B2) . The equations (B1) and (... Let p and q be two fixed non zero integers verifying the condition gcd(p,q) = 1. We check solutions in non zero integers a1,b1,a2,b2 and a3 for the following Diophantine equations: (B1) (B2) . The equations (B1) and (B2) were considered by R.C. Lyndon and J.L. Ullman in [1] and A.F. Beardon in [2] in connection with the freeness of the M?bius group generated by two matrices of namely and where .?They proved that if one of the equations (B1) or (B2) has solutions in non zero integers then the group is not free. We give algorithms to decide if these equations admit solutions. We obtain an arithmetical criteria on p and q for which (B1) admits solutions. We show that for all p and q the equations (B1) and (B2) have only a finite number of solutions. 展开更多
关键词 diophantine equation Mobius Groups Free Group
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Fermat and Pythagoras Divisors for a New Explicit Proof of Fermat’s Theorem:a4 + b4 = c4. Part I
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作者 Prosper Kouadio Kimou François Emmanuel Tanoé Kouassi Vincent Kouakou 《Advances in Pure Mathematics》 2024年第4期303-319,共17页
In this paper we prove in a new way, the well known result, that Fermat’s equation a<sup>4</sup> + b<sup>4</sup> = c<sup>4</sup>, is not solvable in ℕ , when abc≠0 . To show this ... In this paper we prove in a new way, the well known result, that Fermat’s equation a<sup>4</sup> + b<sup>4</sup> = c<sup>4</sup>, is not solvable in ℕ , when abc≠0 . To show this result, it suffices to prove that: ( F 0 ): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is not solvable in ℕ , (where a 1 , b 1 , c 1 ∈2ℕ+1 , pairwise primes, with necessarly 2≤s∈ℕ ). The key idea of our proof is to show that if (F<sub>0</sub>) holds, then there exist α 2 , β 2 , γ 2 ∈2ℕ+1 , such that ( F 1 ): α 2 4 + ( 2 s−1 β 2 ) 4 = γ 2 4 , holds too. From where, one conclude that it is not possible, because if we choose the quantity 2 ≤ s, as minimal in value among all the solutions of ( F 0 ) , then ( α 2 ,2 s−1 β 2 , γ 2 ) is also a solution of Fermat’s type, but with 2≤s−1<s , witch is absurd. To reach such a result, we suppose first that (F<sub>0</sub>) is solvable in ( a 1 ,2 s b 1 , c 1 ) , s ≥ 2 like above;afterwards, proceeding with “Pythagorician divisors”, we creat the notions of “Fermat’s b-absolute divisors”: ( d b , d ′ b ) which it uses hereafter. Then to conclude our proof, we establish the following main theorem: there is an equivalence between (i) and (ii): (i) (F<sub>0</sub>): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is solvable in ℕ , with 2≤s∈ℕ , ( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs. (ii) ∃( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs, for wich: ∃( b ′ 2 , b 2 , b ″ 2 )∈ ( 2ℕ+1 ) 3 coprime in pairs, and 2≤s∈ℕ , checking b 1 = b ′ 2 b 2 b ″ 2 , and such that for notations: S=s−λ( s−1 ) , with λ∈{ 0,1 } defined by c 1 − a 1 2 ≡λ( mod2 ) , d b =gcd( 2 s b 1 , c 1 − a 1 )= 2 S b 2 and d ′ b = 2 s−S b ′ 2 = 2 s B 2 d b , where ( 2 s B 2 ) 2 =gcd( b 1 2 , c 1 2 − a 1 2 ) , the following system is checked: { c 1 − a 1 = d b 4 2 2+λ = 2 2−λ ( 2 S−1 b 2 ) 4 c 1 + a 1 = 2 1+λ d ′ b 4 = 2 1+λ ( 2 s−S b ′ 2 ) 4 c 1 2 + a 1 2 =2 b ″ 2 4;and this system implies: ( b 1−λ,2 4 ) 2 + ( 2 4s−3 b λ,2 4 ) 2 = ( b ″ 2 2 ) 2;where: ( b 1−λ,2 , b λ,2 , b ″ 2 )={ ( b ′ 2 , b 2 , b ″ 2 )  if λ=0 ( b 2 , b ′ 2 , b ″ 2 )  if λ=1;From where, it is quite easy to conclude, following the method explained above, and which thus closes, part I, of this article. . 展开更多
关键词 Factorisation in Greatest Common Divisor Pythagoras equation Pythagorician Triplets Fermat's equations Pythagorician Divisors Fermat's Divisors diophantine equations of Degree 2 4-Integral Closure of in
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Three- and Four-Dimensional Generalized Pythagorean Numbers
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作者 Alfred Wünsche 《Advances in Pure Mathematics》 2024年第1期1-15,共15页
The Pythagorean triples (a, b | c) of planar geometry which satisfy the equation a<sup>2</sup>+b<sup>2</sup>=c<sup>2</sup> with integers (a, b, c) are generalized to 3D-Pythagorean ... The Pythagorean triples (a, b | c) of planar geometry which satisfy the equation a<sup>2</sup>+b<sup>2</sup>=c<sup>2</sup> with integers (a, b, c) are generalized to 3D-Pythagorean quadruples (a, b, c | d) of spatial geometry which satisfy the equation a<sup>2</sup>+b<sup>2</sup>+c<sup>2</sup>=d<sup>2</sup> with integers (a, b, c, d). Rules for a parametrization of the numbers (a, b, c, d) are derived and a list of all possible nonequivalent cases without common divisors up to d<sup>2</sup> is established. The 3D-Pythagorean quadruples are then generalized to 4D-Pythagorean quintuples (a, b, c, d | e) which satisfy the equation a<sup>2</sup>+b<sup>2</sup>+c<sup>2</sup>+d<sup>2</sup>=e<sup>2</sup> and a parametrization is derived. Relations to the 4-square identity are discussed which leads also to the N-dimensional case. The initial 3D- and 4D-Pythagorean numbers are explicitly calculated up to d<sup>2</sup>, respectively, e<sup>2</sup>. 展开更多
关键词 Number Theory Pythagorean Triples Tesseract 4-Square Identity diophantine equation
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The Odd Solutions of Equations Involving Euler-Like Function
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作者 Jiaxin Wu Zhongyan Shen 《Advances in Pure Mathematics》 2021年第5期440-446,共7页
<span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><spa... <span style="white-space:nowrap;"><em><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">φ</span></span></span></span></span></span><sub>e</sub></em>(<em>n</em>) </span>is a function similar to Euler function <em><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">φ</span></span></span></span></span></em>(<em>n</em>). We discussed and obtained all the odd solutions of the equations <em style="white-space:normal;"><span style="white-space:nowrap;">φ<sub>e</sub></span></em><span style="white-space:nowrap;">(<em>xy</em>) </span><span style="white-space:nowrap;">= <em style="white-space:normal;"><span style="white-space:nowrap;">φ<sub>e</sub></span></em><span style="white-space:nowrap;"><sub></sub>(<em>x</em>)</span></span><span style="white-space:nowrap;"> + </span><span style="white-space:nowrap;">2</span><em style="white-space:normal;"><span style="white-space:nowrap;"><em style="white-space:normal;"><span style="white-space:nowrap;">φ<sub>e</sub></span></em></span></em><span style="white-space:nowrap;"><span style="white-space:nowrap;"><sub></sub>(</span></span><em style="white-space:normal;"><span style="white-space:nowrap;"><em style="white-space:normal;"><span style="white-space:nowrap;">y)</span></em></span></em>, <em style="white-space:normal;"><span style="white-space:nowrap;">φ<sub>e</sub></span></em>(<em>xy</em>) = 2<em style="white-space:normal;"><span style="white-space:nowrap;">φ<sub>e</sub></span></em><sub></sub>(<em>x</em>) + 3<em style="white-space:normal;"><span style="white-space:nowrap;"><em style="white-space:normal;"><span style="white-space:nowrap;">φ<sub>e</sub></span></em></span></em><sub></sub>(<em style="white-space:normal;"><span style="white-space:nowrap;">y) </span></em>and <em style="white-space:normal;"><span style="white-space:nowrap;">φ<sub>e</sub></span></em><span style="white-space:normal;">(</span><em style="white-space:normal;">xyz</em><span style="white-space:normal;">) = </span><em style="white-space:normal;"><span style="white-space:nowrap;">φ<sub>e</sub></span></em><sub style="white-space:normal;"></sub><span style="white-space:normal;">(</span><em style="white-space:normal;">x</em><span style="white-space:normal;">) + </span><em style="white-space:normal;"><span style="white-space:nowrap;"><em style="white-space:normal;"><span style="white-space:nowrap;">φ<sub>e</sub></span></em></span></em><sub style="white-space:normal;"></sub><span style="white-space:normal;">(</span><em style="white-space:normal;"><span style="white-space:nowrap;">y)</span></em> <span style="white-space:normal;">+ </span><em style="white-space:normal;"><span style="white-space:nowrap;"><em style="white-space:normal;"><span style="white-space:nowrap;">φ<sub>e</sub></span></em></span></em><sub style="white-space:normal;"></sub><span style="white-space:normal;">(</span><em style="white-space:normal;"><span style="white-space:nowrap;">z)</span></em><span style="white-space:normal;"> </span>by using the methods and techniques of elementary number theory. 展开更多
关键词 Euler-Like Function diophantine equation Odd Solutions
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On Ellipsoids Attached to Root Systems
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作者 Anatoli Loutsiouk 《Journal of Applied Mathematics and Physics》 2016年第8期1513-1521,共9页
For any finite-dimensional complex semisimple Lie algebra, two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are Diophantine equations, and the Weyl group acts on the sets of all... For any finite-dimensional complex semisimple Lie algebra, two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are Diophantine equations, and the Weyl group acts on the sets of all their Diophantine solutions. This provides two realizations (primary and secondary) of the Weyl group on the sets of Diophantine solutions of the equations of the ellipsoids. The primary realization of the Weyl group suggests an order on the Weyl group, which is stronger than the Chevalley-Bruhat ordering of the Weyl group, and which provides an algorithm for the Chevalley-Bruhat ordering. The secondary realization of the Weyl group provides an algorithm for constructing all reduced expressions for any of its elements, and thus provides another way for the Chevalley-Bruhat ordering of the Weyl group. 展开更多
关键词 Complex Semisimple Lie Algebra Cartan Subalgebra Weyl Group Cartan Matrix Primary and Secondary Ellipsoids diophantine equations Geometric Realizations Coxeter Relations Dynkin Diagram Chevalley-Bruhat Ordering Reduced Expressions
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Relating Optimization Problems to Systems of Inequalities and Equalities
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作者 H. W. Corley E. O. Dwobeng 《American Journal of Operations Research》 2020年第6期284-298,共15页
In quantitative decision analysis, an analyst applies mathematical models to make decisions. Frequently these models involve an optimization problem to determine the values of the decision variables, a system </spa... In quantitative decision analysis, an analyst applies mathematical models to make decisions. Frequently these models involve an optimization problem to determine the values of the decision variables, a system </span><i><span style="font-family:Verdana;">S</span></i><span style="font-family:Verdana;"> of possibly non</span></span><span style="font-family:Verdana;">- </span><span style="font-family:Verdana;">li</span><span style="font-family:Verdana;">near inequalities and equalities to restrict these variables, or both. In this</span><span style="font-family:""><span style="font-family:Verdana;"> note, </span><span style="font-family:Verdana;">we relate a general nonlinear programming problem to such a system </span><i><span style="font-family:Verdana;">S</span></i><span style="font-family:Verdana;"> in</span><span style="font-family:Verdana;"> such </span><span style="font-family:Verdana;">a way as to provide a solution of either by solving the other—with certain l</span><span style="font-family:Verdana;">imitations. We first start with </span><i><span style="font-family:Verdana;">S</span></i><span style="font-family:Verdana;"> and generalize phase 1 of the two-phase simplex method to either solve </span><i><span style="font-family:Verdana;">S</span></i><span style="font-family:Verdana;"> or establish that a solution does not exist. A conclusion is reached by trying to solve </span><i><span style="font-family:Verdana;">S</span></i><span style="font-family:Verdana;"> by minimizing a sum of artificial variables subject to the system </span><i><span style="font-family:Verdana;">S</span></i><span style="font-family:Verdana;"> as constraints. Using examples, we illustrate </span><span style="font-family:Verdana;">how this approach can give the core of a cooperative game and an equili</span><span style="font-family:Verdana;">brium for a noncooperative game, as well as solve both linear and nonlinear goal programming problems. Similarly, we start with a general nonlinear programming problem and present an algorithm to solve it as a series of systems </span><i><span style="font-family:Verdana;">S</span></i><span style="font-family:Verdana;"> by generalizing the </span></span><span style="font-family:Verdana;">“</span><span style="font-family:Verdana;">sliding objective</span><span style="font-family:Verdana;"> function </span><span style="font-family:Verdana;">method</span><span style="font-family:Verdana;">”</span><span style="font-family:Verdana;"> for</span><span style="font-family:Verdana;"> two-dimensional linear programming. An example is presented to illustrate the geometrical nature of this approach. 展开更多
关键词 OPTIMIZATION Inequalities and Equalities Goal Programming GAMES diophantine equations
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Computational Resolution of a Boolean Equation of 21 Variables
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作者 Esther Claudine Bitye Mvondo 《American Journal of Operations Research》 2022年第5期157-178,共22页
The Alienor method has been elaborated at the beginning of the 1980s by Yves Cherruault and Arthur Guillez (1983). The following people have also greatly contributed to the improvement of this new optimization method:... The Alienor method has been elaborated at the beginning of the 1980s by Yves Cherruault and Arthur Guillez (1983). The following people have also greatly contributed to the improvement of this new optimization method: Blaise Somé, Gaspar Mora, Balira Konfé, Jean Claude Mazza and Esther Claudine Bityé Mvondo. The basic idea consists in using a reducing transformation allowing us to simplify a multivariable optimization problem to a new optimization problem according to a single variable. The rational gestion of enterprises leads generally to the use of Operational Research, often called management science. The term Operational Research means a scientific approach to decision making, that seeks optimization in a system. Consequently, it is better to take the right decisions. Otherwise, fatal consequences can occur instantaneously [1]. Generally, we have to maximize the global profit margin, taking into account some constraints. For instance, in an integer programming problem, some or all the variables are required to be nonnegative integers. In this paper, we present new reducing transformations for global optimization in integer, binary and mixed variables as well as the applications in Boolean algebra by solving a Boolean Equation of 21 variables. The applications in Operational Research are presented on various examples, resolved by using the tabulator Excel of Microsoft. 展开更多
关键词 Alpha-Dense Curves Boolean equations diophantine equations Global Optimization and Operational Research
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On the solutions of a system of two Diophantine equations 被引量:4
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作者 LUO JiaGui YUAN PingZhi 《Science China Mathematics》 SCIE 2014年第7期1401-1418,共18页
We obtain all positive integer solutions(m1,m2,a,b) with a &gt; b,gcd(a,b) = 1 to the system of Diophantine equations km21- lat1bt2a2r= C1,km22- lat1bt2b2r= C2,with C1,C2 ∈ {-1,1,-2,2,-4,4},and k,l,t1,t2,r ∈ Z ... We obtain all positive integer solutions(m1,m2,a,b) with a &gt; b,gcd(a,b) = 1 to the system of Diophantine equations km21- lat1bt2a2r= C1,km22- lat1bt2b2r= C2,with C1,C2 ∈ {-1,1,-2,2,-4,4},and k,l,t1,t2,r ∈ Z such that k &gt; 0,l &gt; 0,r &gt; 0,t1 &gt; 0,t2 0,gcd(k,l) = 1,and k is square-free. 展开更多
关键词 minimal solution fundamental solution Jacobi symbol diophantine equation
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Inductive Rings and Systems of Diophantine Equations
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作者 Rong Fang BIE Shi Qiang WANG 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2006年第5期1549-1556,共8页
In this paper, by using model-theoretic methods, it is shown that some systems of unsolved cubic diophantine equations in number theory can have solutions in certain inductive extension rings of the ring I of rational... In this paper, by using model-theoretic methods, it is shown that some systems of unsolved cubic diophantine equations in number theory can have solutions in certain inductive extension rings of the ring I of rational integers. These inductive rings are not fields, and every element of them is a sum of 4 cubes and a sum of 3 squares. Also some of them satisfy the Goldbach conjecture and some others don't. 展开更多
关键词 Inductive rings systems of unsolved cubic diophantine equations Model theory
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