Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely...Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely.The aim of this paper is to study the similar problems about Fermat’s Last Theorem for multivariate(skew)-polynomials with any characteristic.展开更多
Let p be a prime with p≡3(mod 4). In this paper,by using some results relate the representation of integers by primitive binary quadratic forms,we prove that if x,y,z are positive integers satisfying x^p+y^p=z^p, p|x...Let p be a prime with p≡3(mod 4). In this paper,by using some results relate the representation of integers by primitive binary quadratic forms,we prove that if x,y,z are positive integers satisfying x^p+y^p=z^p, p|xyz, x<y<z, then y>p^(6p-2)/2.展开更多
In this paper, some conmments on the proof of Fermat’s last theorem are proposed.The main resuilt is thai the proof proposed by Wong Chiahe is only part of proof for fermat’s last theorem. That is to sqy ,the proof...In this paper, some conmments on the proof of Fermat’s last theorem are proposed.The main resuilt is thai the proof proposed by Wong Chiahe is only part of proof for fermat’s last theorem. That is to sqy ,the proof is not all-full proof to Fermat’s last theorem.展开更多
i) Instead of x￣n+ y￣n = z￣n ,we use as the general equation of Fermat's Last Theorem (FLT),where a and b are two arbitrary natural numbers .By means of binomial expansion ,(0.1) an be written as Because a￣...i) Instead of x￣n+ y￣n = z￣n ,we use as the general equation of Fermat's Last Theorem (FLT),where a and b are two arbitrary natural numbers .By means of binomial expansion ,(0.1) an be written as Because a￣r-(-b)￣r always contains a +b as its factor ,(0.2) can be written as where φ_r =[a￣r-(-b)￣r]/ (a+b ) are integers for r=1 . 2, 3. ...n (ii) Lets be a factor of a+b and let (a +b) = se. We can use x= sy to transform (0.3 ) to the following (0.4)(iii ) Dividing (0.4) by s￣2 we have On the left side of (0.5) there is a polynomial of y with integer coefficient and on the right side there is a constant cφ/s .If cφ/s is not an integer ,then we cannot find an integer y to satisfy (0.5), and then FLT is true for this case. If cφ_n/s is an integer ,we may change a and c such the cφ_n/s≠an integer .展开更多
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
A new method based on Huygens' principle and Fermat's principle is presented to calculate seismic traveltime and ray-paths in this paper. The algorithm can be easily understood and programmed, and can be appli...A new method based on Huygens' principle and Fermat's principle is presented to calculate seismic traveltime and ray-paths in this paper. The algorithm can be easily understood and programmed, and can be applied to heteroge neous media in order to eliminate its disadvantage of slowness, we have improved the basic algorithm to speed its calculation to practical stage without the loss of its accuracy.展开更多
Traveltime tomography is a technique to reconstruct acoustic, seismic, or electromagnetic wave-speed distributions from first arrival traveltime data. The ray paths that should be used for tomographic techniques stro...Traveltime tomography is a technique to reconstruct acoustic, seismic, or electromagnetic wave-speed distributions from first arrival traveltime data. The ray paths that should be used for tomographic techniques strongly depend on the wave-speed distribution. In this paper, a new method is proposed for finding out the ray paths from Fermat's principle, that means the traveltime of the ray path should be a minimum value. The problem of finding out the ray path is actually an optimum problem. Our new method uses the idea to find out the shortest path in a weighted directed graph to solve the problem. The ray paths found out by this method are used in the iterative reconstruction algorithm. Computer simulation result produced by this reconstruction algorithm is better than that by the conventional ones. It also shows that the new algorithm is effective with good convergency and stability.展开更多
基金supported by the National Natural Science Foundation of China(12131015,12071422).
文摘Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely.The aim of this paper is to study the similar problems about Fermat’s Last Theorem for multivariate(skew)-polynomials with any characteristic.
文摘Let p be a prime with p≡3(mod 4). In this paper,by using some results relate the representation of integers by primitive binary quadratic forms,we prove that if x,y,z are positive integers satisfying x^p+y^p=z^p, p|xyz, x<y<z, then y>p^(6p-2)/2.
文摘In this paper, some conmments on the proof of Fermat’s last theorem are proposed.The main resuilt is thai the proof proposed by Wong Chiahe is only part of proof for fermat’s last theorem. That is to sqy ,the proof is not all-full proof to Fermat’s last theorem.
文摘i) Instead of x￣n+ y￣n = z￣n ,we use as the general equation of Fermat's Last Theorem (FLT),where a and b are two arbitrary natural numbers .By means of binomial expansion ,(0.1) an be written as Because a￣r-(-b)￣r always contains a +b as its factor ,(0.2) can be written as where φ_r =[a￣r-(-b)￣r]/ (a+b ) are integers for r=1 . 2, 3. ...n (ii) Lets be a factor of a+b and let (a +b) = se. We can use x= sy to transform (0.3 ) to the following (0.4)(iii ) Dividing (0.4) by s￣2 we have On the left side of (0.5) there is a polynomial of y with integer coefficient and on the right side there is a constant cφ/s .If cφ/s is not an integer ,then we cannot find an integer y to satisfy (0.5), and then FLT is true for this case. If cφ_n/s is an integer ,we may change a and c such the cφ_n/s≠an integer .
文摘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. .
文摘A new method based on Huygens' principle and Fermat's principle is presented to calculate seismic traveltime and ray-paths in this paper. The algorithm can be easily understood and programmed, and can be applied to heteroge neous media in order to eliminate its disadvantage of slowness, we have improved the basic algorithm to speed its calculation to practical stage without the loss of its accuracy.
文摘Traveltime tomography is a technique to reconstruct acoustic, seismic, or electromagnetic wave-speed distributions from first arrival traveltime data. The ray paths that should be used for tomographic techniques strongly depend on the wave-speed distribution. In this paper, a new method is proposed for finding out the ray paths from Fermat's principle, that means the traveltime of the ray path should be a minimum value. The problem of finding out the ray path is actually an optimum problem. Our new method uses the idea to find out the shortest path in a weighted directed graph to solve the problem. The ray paths found out by this method are used in the iterative reconstruction algorithm. Computer simulation result produced by this reconstruction algorithm is better than that by the conventional ones. It also shows that the new algorithm is effective with good convergency and stability.
