Products of Odd Numbers or Prime Number Can Generate the Three Members’ Families of Fermat Last Theorem and the Theorem Is Valid for Summation of Squares of More Than Two Natural Numbers

Fermat’s last theorem, had the statement that there are no natural numbers A, B, and C such that An + Bn = Cn, in which n is a natural number greater than 2. We have shown that any product of two odd numbers can generate Fermat or Pythagoras triple (A, B, C) following n = 2 and also it is applicable A2 + B2 + C2 + D2 + so on =An2 where all are natural numbers.

Five Consecutive Positive Odd Numbers None of Which Can Be Expressed as a Sum of Two Prime Powers Ⅱ

In this paper, we find two integers k0, m of 159 decimal digits such that if k ≡ k0 （mod m）, then none of five consecutive odd numbers k, k - 2, k - 4, k - 6 and k - 8 can be expressed in the form 2^n ± p^α, where p is a prime and n, α are nonnegative integers.

Very Original Proofs of Two Famous Problems: “Are There Any Odd Perfect Numbers?” (Unsolved until to Date) and “Fermat’s Last Theorem: A New Proof of Theorem (Less than One and a Half Pages) and Its Generalization”

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. Note that the second solution is very short and does not exceed one and a half pages. 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. 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.

Generalized Central Factorial Numbers with Odd Arguments

In this paper, we consider r-generalization of the central factorial numbers with odd arguments of the first and second kind. Mainly, we obtain various identities and properties related to these numbers. Matrix representation and the relation between these numbers and Pascal matrix are given. Furthermore, the distributions of the signless r-central factorial numbers are derived. In addition, connections between these numbers and the Legendre-Stirling numbers are given.

Some Results of a Certain Odd Perfect Numb er

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.

A survey on delayed feedback control of chaos

This paper introduces the basic idea and provides the mathematical formulation of the delayed feedback control （DFC） methodology, which has been widely used in chaos control. Stability analysis including the well-known odd number linfitation of the DFC is reviewed. Some new developments in characterizing the limitation of the DFC are presented. Various modified DFC methods, which are developed in order to overcome the odd number limitation, are also described. Finally, some open problems in this research field are discussed.

Trapping the Short-Chain Odd Carbon Number Olefins Using Nickel(Ⅱ)-Catalyzed Tandem Ethylene Oligomerization and Friedel-Crafts Alkylation of Toluene

Nickel(Ⅱ)complexes with pyrazole-based ligands are widely employed in catalysis of ethylene oligomerization and subsequent Friedel-Crafts alkylation of toluene.We have prepared ten new nickel(Ⅱ)dibromide complexes with various substituted bis(azolyl)methanes.They have been characterized using^(1)H NMR,IR,high resolution mass spectrometry and elemental analysis.The structures of three complexes have been unambiguously established using X-ray diffraction.It was found that these complexes in the presence of Et2AlCl or Et_(3)Al_(2)Cl_(3)are active both in ethylene oligomerization and Friedel-Crafts alkylation processes(activity up to 3720 kgoligomer·mol[Ni]^(−1)·h^(−1)).The use of Et_(3)Al_(2)Cl_(3)results in a higher share of alkylated products(up to 60%).Moreover,catalytic systems activated with Et_(3)Al_(2)Cl_(3)produced small amounts of odd carbon number olefins(up to 0.8%).The Friedel-Crafts alkylation was used as a trap for previously undetected short-chain odd carbon number olefins(C_(3)and C_(5)).