The notion of weakly relatively prime and W-Gr6bner basis in K[x1, x2,…, xn] are given. The following results are obtained： for polynomials fl, f2, ..., fm, {f1^λ1, f2^λ2,…, fm^λm} is a GrSbner basis if and only...The notion of weakly relatively prime and W-Gr6bner basis in K[x1, x2,…, xn] are given. The following results are obtained： for polynomials fl, f2, ..., fm, {f1^λ1, f2^λ2,…, fm^λm} is a GrSbner basis if and only if f1, f2, …, fm are pairwise weakly relatively prime with λ1, λ2, …, λm arbitrary non-negative integers; polynomial composition by θ = （θ1,θ2, …, θn） commutes with monomial-Grobner bases computation if and only if θ1, θ2, , θm are pairwise weakly relatively prime.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, 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
For any integer s ≥ 2, let μs be the least integer so that every integer l 〉 μs is the sum of exactly s integers which are pairwise relatively prime. In 1964, Sierpifiski asked for the determination of μs. Let Pi...For any integer s ≥ 2, let μs be the least integer so that every integer l 〉 μs is the sum of exactly s integers which are pairwise relatively prime. In 1964, Sierpifiski asked for the determination of μs. Let Pi be the i-th prime and let μs = p2 ＋P3 ＋ … ＋ps＋1＋ cs. Recently, the authors solved this problem. In particular, we have （1） cs = -2 if and only if s = 2; （2） the set of integers s with cs= 1100 has asymptotic density one; （3） cs ∈ A for all s ≥ 3, where A is an explicit set with A [2, 1100] and ｜A｜ = 125. In this paper, we prove that, （1） for every a ∈ A, there exists an index s with cs = there are infinitely many s with es = a. We also point out can be applied to this problem. a; （2） under Dickson＇s conjecture, for every a∈ A, that recent progress on small gaps between primes展开更多
基金Supported by the NSFC (10771058, 11071062, 10871205), NSFH (10JJ3065)Scientific Research Fund of Hunan Provincial Education Department (10A033)Hunan Provincial Degree and Education of Graduate Student Foundation (JG2009A017)
文摘The notion of weakly relatively prime and W-Gr6bner basis in K[x1, x2,…, xn] are given. The following results are obtained： for polynomials fl, f2, ..., fm, {f1^λ1, f2^λ2,…, fm^λm} is a GrSbner basis if and only if f1, f2, …, fm are pairwise weakly relatively prime with λ1, λ2, …, λm arbitrary non-negative integers; polynomial composition by θ = （θ1,θ2, …, θn） commutes with monomial-Grobner bases computation if and only if θ1, θ2, , θm are pairwise weakly relatively prime.
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, 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.
基金supported by National Natural Science Foundation of China(Grant Nos.11371195 and 11201237)
文摘For any integer s ≥ 2, let μs be the least integer so that every integer l 〉 μs is the sum of exactly s integers which are pairwise relatively prime. In 1964, Sierpifiski asked for the determination of μs. Let Pi be the i-th prime and let μs = p2 ＋P3 ＋ … ＋ps＋1＋ cs. Recently, the authors solved this problem. In particular, we have （1） cs = -2 if and only if s = 2; （2） the set of integers s with cs= 1100 has asymptotic density one; （3） cs ∈ A for all s ≥ 3, where A is an explicit set with A [2, 1100] and ｜A｜ = 125. In this paper, we prove that, （1） for every a ∈ A, there exists an index s with cs = there are infinitely many s with es = a. We also point out can be applied to this problem. a; （2） under Dickson＇s conjecture, for every a∈ A, that recent progress on small gaps between primes
