This article presents very original and relatively brief or very brief proofs about of two famous problems: 1) Are there any odd perfect numbers? and 2) “Fermat’s last theorem: A new proof of theorem and its general...This article presents very original and relatively brief or very brief proofs about of two famous problems: 1) Are there any odd perfect numbers? and 2) “Fermat’s last theorem: A new proof of theorem and its generalization”. They are achieved with elementary mathematics. This is why these proofs can be easily understood by any mathematician or anyone who knows basic mathematics. Note that, in both problems, proof by contradiction was used as a method of proof. The first of the two problems to date has not been resolved. Its proof is completely original and was not based on the work of other researchers. On the contrary, it was based on a simple observation that all natural divisors of a positive integer appear in pairs. The aim of the first work is to solve one of the unsolved, for many years, problems of the mathematics which belong to the field of number theory. I believe that if the present proof is recognized by the mathematical community, it may signal a different way of solving unsolved problems. For the second problem, it is very important the fact that it is generalized to an arbitrarily large number of variables. This generalization is essentially a new theorem in the field of the number theory. To the classical problem, two solutions are given, which are presented in the chronological order in which they were achieved. <em>Note that the second solution is very short and does not exceed one and a half pages</em>. This leads me to believe that Fermat, as a great mathematician was not lying and that he had probably solved the problem, as he stated in his historic its letter, with a correspondingly brief solution. <em>To win the bet on the question of whether Fermat was telling truth or lying, go immediately to the end of this article before the General Conclusions.</em>展开更多
Let n be a positive integer satisfying n >1; ω(n) denotes the number of distinct prime factors of n ; σ(n) denotes the sum of the positive divisors of n . If σ(n)=2n then n is said to be a perfect number and if ...Let n be a positive integer satisfying n >1; ω(n) denotes the number of distinct prime factors of n ; σ(n) denotes the sum of the positive divisors of n . If σ(n)=2n then n is said to be a perfect number and if σ(n)=kn(k≥3) then n is said to be a multiply perfect number. In this paper according to Euler theorem and Fermat theorem, we discuss the result of σ(n)=ω(n)n and prove that only n=2 3·3·5, 2 5·3·7, 2 5·3 3·5·7 satisfies σ(n)= ω(n) n(ω(n)≥3). ...展开更多
Define the total number of distinct prime factors of an odd perfect number n asω(n). We prove that if n is an odd perfect number which is relatively prime to 3 and 5 and7, then ω(n) ≥ 107. And using this result, we...Define the total number of distinct prime factors of an odd perfect number n asω(n). We prove that if n is an odd perfect number which is relatively prime to 3 and 5 and7, then ω(n) ≥ 107. And using this result, we give a conclusion that the third largest prime factor of such an odd perfect number exceeds 1283.展开更多
文摘This article presents very original and relatively brief or very brief proofs about of two famous problems: 1) Are there any odd perfect numbers? and 2) “Fermat’s last theorem: A new proof of theorem and its generalization”. They are achieved with elementary mathematics. This is why these proofs can be easily understood by any mathematician or anyone who knows basic mathematics. Note that, in both problems, proof by contradiction was used as a method of proof. The first of the two problems to date has not been resolved. Its proof is completely original and was not based on the work of other researchers. On the contrary, it was based on a simple observation that all natural divisors of a positive integer appear in pairs. The aim of the first work is to solve one of the unsolved, for many years, problems of the mathematics which belong to the field of number theory. I believe that if the present proof is recognized by the mathematical community, it may signal a different way of solving unsolved problems. For the second problem, it is very important the fact that it is generalized to an arbitrarily large number of variables. This generalization is essentially a new theorem in the field of the number theory. To the classical problem, two solutions are given, which are presented in the chronological order in which they were achieved. <em>Note that the second solution is very short and does not exceed one and a half pages</em>. This leads me to believe that Fermat, as a great mathematician was not lying and that he had probably solved the problem, as he stated in his historic its letter, with a correspondingly brief solution. <em>To win the bet on the question of whether Fermat was telling truth or lying, go immediately to the end of this article before the General Conclusions.</em>
文摘Let n be a positive integer satisfying n >1; ω(n) denotes the number of distinct prime factors of n ; σ(n) denotes the sum of the positive divisors of n . If σ(n)=2n then n is said to be a perfect number and if σ(n)=kn(k≥3) then n is said to be a multiply perfect number. In this paper according to Euler theorem and Fermat theorem, we discuss the result of σ(n)=ω(n)n and prove that only n=2 3·3·5, 2 5·3·7, 2 5·3 3·5·7 satisfies σ(n)= ω(n) n(ω(n)≥3). ...
基金Foundation item： Supported by the Science Foundation of Kashgar Teacher＇s College（112390）
文摘Define the total number of distinct prime factors of an odd perfect number n asω(n). We prove that if n is an odd perfect number which is relatively prime to 3 and 5 and7, then ω(n) ≥ 107. And using this result, we give a conclusion that the third largest prime factor of such an odd perfect number exceeds 1283.
