Tree of Fermat-Pramanik Series and Solution of AM +B2 =C2 with Integers Produces a New Series of (C12- B12)=(C22- B22)=(C32- B32)=Others

The Fermat–Pramanik series are like below: .The mathematical principle has been established by factorization principle. The Fermat-Pramanik tree can be grown. It produces branched Fermat-Pramanik series using same principle making Fermat-Pramanik chain. Branched chain can be propagated at any point of the main chain with indefinite length using factorization principle as follows: Same principle is applicable for integer solutions of AM+B2=C2which produces series of the type . It has been shown that this equation is solvable with N{A, B, C, M}. where , , M=M1+M2 and M1>M2. Subsequently, it has been shown that using M= M1+M2+M3+... The combinations of Ms should be taken so that the values of both the parts (Cn+Bn) and (Cn-Bn) should be even or odd for obtaining Z{B,C}. Hence, it has been shown that the Fermat triple can generate a) Fermat-Pramanik multiplate, b) Fermat-Pramanik Branched multiplate and c) Fermat-Pramanik deductive series. All these formalisms are useful for development of new principle of cryptography. .

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.

Nitric oxide mediated Staphylococcus aureus pathogenesis and protective role of nanoconjugated vancomycin

Objective:To investigate the relation between fruit seeds,plants residuals and appendicitis. Methods:Among cases that underwent appendectomy,the appendicitis cases having fruit seeds and undigested plant residuals in their etiology were examined retrospectively.Also, histopathological features,age,sex,and parameters of morbidity and mortality were used. Results:Fruit seed was found in one case(0.05%) with presence of pus in appendix lumen, undigested plant residuals in 7 cases(0.35%).It was determined that there were appendix inflammation in 2 of the plant residuals cases,while there were obstruction and lymphoid hyperplasia in the appendix lumen of 5 cases.No mortality was observed.Conclusions:The ratio of acute appendicitis caused by plants is minimal among all appendectomised patients, but avoidence of eating undigested fruit seeds and chewing plants well may help to prevent appendicitis.

Antioxidative effect of folate-modified chitosan nanoparticles

Objective:To evaluate the potency of carboxymethyl chitosan-2,2' ethylenedioxy bisethylamine-folate(CMC-EDBE-FA) on tissue injury,antioxidant status and glutathione system in tissue mitochondria and serum against nicotine-induced oxidative stress in mice.Methods: CMC-EDBE-FA was prepared on basis of carboxymethyl chitosan tagged with folic acid by covalently linkage through 2,2' ethylenedioxy bis-ethylamine.Animals were divided into four groups,i.e.,control,nicotine(1 mg/kg bw/day),CMC-EDBE-FA(1 mg/kg bw/day) and nicotine(1 mg/kg bw/day) and CMC-EDBE-FA(1 mg/kg bw/day) for 7 days.Levels of lipid peroxidation, oxidized glutathione level,antioxidant enzyme status and DNA damage were observed and compared.Results:The significantly increase of lipid peroxidation,oxidized glutathione levels and DNA damage was observed in nicotine treated group as compared with control group;those were significantly reduced in CMC-EDBE-FA supplemented group.Moreover,significantly reduced antioxidant status in nicotine treated group was effectively ameliorated by the supplementation of CMC-EDBE-FA.Only CMC-EDBE-FA treated groups showed no significant change as compared with control group;rather than it repairs the tissue damage of nicotine treated group.Conclusions:These findings suggest that CMC-EDBE-FA is non-toxic and ameliorates nicotine-induced toxicity.

Biocompatibility of folate-modified chitosan nanoparticles

Objective:To evaluate the acute toxicity of carboxymethyl chitosan-2,2' ethylenedioxy bisethylamine-folate(CMC-EDBE-FA) and as well as possible effect on microbial growth and in vitro cell cyto-toxicity.Methods:CMC-EDBE-FA was prepared on basis of carboxymethyl chitosan tagged with folic acid by covalently linkage through 2,2' elhylenedioxy bis-ethylamine. In vivo acute toxicity,in vitro cyto-toxicity and antimicrobial activity of CMC-EDBE-FA nanoparticle were determined.Results:Vancomycin exhibited the antibacterial activity against vancomycin sensitive Staphylococcus aureus,but CMC-EDBE-FA nanoparticle did not give any antibacterial activity as evidenced by minimal inhibitory concentration(MIC),minimal bactericidal concentration(MBC),disc agar diffusion(DAD) and killing kinetic assay.Further,the CMC-EDBE-FA nanoparticle showed no signs of in vivo acute toxicity up to a dose level of 1000 mg/kg p.o.,and as well as in vitro cyto-toxicity up to 230 μg/mL.Conclnsions:These findings suggest that CMC-EDBE-FA nanoparticle is expected to be safe for biomedical applications.