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Automorphic Functions And Fermat’s Last Theorem (1)

Chun-Xuan Jiang

P.O. Box 3924, Beijing 100854, China

jiangchunxuan@vip.sohu.com

Abstract: In 1637 Fermat wrote: “It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into powers of like degree: I have discovered a truly marvelous proof, which this margin is too small to contain.” This means:  has no integer solutions, all different from 0 (i.e., it has only the trivial solution, where one of the integers is equal to 0). It has been called Fermat’s last theorem (FLT). It suffices to prove FLT for exponent 4. and every prime exponent . Fermat proved FLT for exponent 4. Euler proved FLT for exponent 3. In this paper using automorphic functions we prove FLT for exponents  and , where  is an odd prime. The proof of FLT must be direct. But indirect proof of FLT is disbelieving.

[Chun-Xuan Jiang. Automorphic Functions And Fermat’s Last Theorem (1). Rep Opinion 2012;4(8):1-6]. (ISSN: 1553-9873). http://www.sciencepub.net/report. 1

doi:10.7537/marsroj040812.01

Keywords: automorphic function; cyclotomic field; Fermat; theorem

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Automorphic Functions And Fermat’s Last Theorem（2）

Chun-Xuan Jiang

P. O. Box 3924, Beijing 100854, P. R. China

Jiangchunxuan@vip.sohu.com

Abstract: In 1637 Fermat wrote: “It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into powers of like degree: I have discovered a truly marvelous proof, which this margin is too small to contain.” This means:  has no integer solutions, all different from 0(i.e., it has only the trivial solution, where one of the integers is equal to 0). It has been called Fermat’s last theorem (FLT). It suffices to prove FLT for exponent 4. and every prime exponent . Fermat proved FLT for exponent 4. Euler proved FLT for exponent 3. In this paper using automorphic functions we prove FLT for exponents  and , where  is an odd prime. The proof of FLT must be direct .But indirect proof of FLT is disbelieving.

[Chun-Xuan Jiang. Automorphic Functions And Fermat’s Last Theorem（2）. Rep Opinion 2012;4(8):7-10]. (ISSN: 1553-9873). http://www.sciencepub.net/report. 2

doi:10.7537/marsroj040812.02

Keywords: automorphic function; cyclotomic field; Fermat; theorem

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Automorphic Functions And Fermat’s Last Theorem（3）(Fermat’s Proof of FLT)

Chun-Xuan Jiang

P. O. Box 3924, Beijing 100854, P. R. China

jiangchunxuan@sohu.com

Abstract: In 1637 Fermat wrote: “It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into powers of like degree: I have discovered a truly marvelous proof, which this margin is too small to contain.” This means:  has no integer solutions, all different from 0(i.e., it has only the trivial solution, where one of the integers is equal to 0). It has been called Fermat’s last theorem (FLT). It suffices to prove FLT for exponent 4 and every prime exponent . Fermat proved FLT for exponent 4. Euler proved FLT for exponent 3. In this paper using automorphic functions we prove FLT for exponents  and , where  is an odd prime. We rediscover the Fermat proof. The proof of FLT must be direct. But indirect proof of FLT is disbelieving.

[Chun-Xuan Jiang. Automorphic Functions And Fermat’s Last Theorem（3）(Fermat’s Proof of FLT). Rep Opinion 2012;4(8):11-14]. (ISSN: 1553-9873). http://www.sciencepub.net/report. 3

doi:10.7537/marsroj040812.03

Keywords: automorphic function; cyclotomic field; Fermat; theorem

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Aotomorphic Functions And Fermat’s Last Theorem(4)

Chun-Xuan Jiang

P. O. Box 3924, Beijing 100854, P. R. China

jiangchunxuan@sohu.com

Abstract: 1637 Fermat wrote: “It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into powers of like degree: I have discovered a truly marvelous proof, which this margin is too small to contain.” This means:  has no integer solutions, all different from 0(i.e., it has only the trivial solution, where one of the integers is equal to 0). It has been called Fermat’s last theorem (FLT). It suffices to prove FLT for exponent 4. and every prime exponent . Fermat proved FLT for exponent 4. Euler proved FLT for exponent 3. In this paper using automorphic functions we prove FLT for exponents  and , where  is an odd prime. We find the Fermat proof. The proof of FLT must be direct. But indirect proof of FLT is disbelieving.

[Chun-Xuan Jiang. Aotomorphic Functions And Fermat’s Last Theorem(4). Rep Opinion 2012;4(8):15-20]. (ISSN: 1553-9873). http://www.sciencepub.net/report. 4

doi:10.7537/marsroj040812.04

Keywords: automorphic function; cyclotomic field; Fermat; theorem

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Automorphic Functions And Fermat’s Last Theorem（5）

Chun-Xuan Jiang

P. O. Box 3924, Beijing 100854, P. R. China

jiangchunxuan@sohu.com

Abstract: In 1637 Fermat wrote: “It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into powers of like degree: I have discovered a truly marvelous proof, which this margin is too small to contain.” This means:  has no integer solutions, all different from 0(i.e., it has only the trivial solution, where one of the integers is equal to 0). It has been called Fermat’s last theorem (FLT). It suffices to prove FLT for exponent 4 and every prime exponent . Fermat proved FLT for exponent 4. Euler proved FLT for exponent 3[8]. In this paper using automorphic functions we prove FLT for exponents  and , where  is an odd prime. The proof of FLT must be direct. But indirect proof of FLT is disbelieving.

[Chun-Xuan Jiang. Automorphic Functions And Fermat’s Last Theorem（5）. Rep Opinion 2012;4(8):21-24]. (ISSN: 1553-9873). http://www.sciencepub.net/report. 5

doi:10.7537/marsroj040812.05

Keywords: automorphic function; cyclotomic field; Fermat; theorem

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Automorphic Functions And Fermat’s Last Theorem（6）

Chun-Xuan Jiang

P. O. Box 3924, Beijing 100854, P. R. China

jiangchunxuan@sohu.com

Abstract: In 1637 Fermat wrote: “It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into powers of like degree: I have discovered a truly marvelous proof, which this margin is too small to contain.” This means:  has no integer solutions, all different from 0(i.e., it has only the trivial solution, where one of the integers is equal to 0). It has been called Fermat’s last theorem (FLT). It suffices to prove FLT for exponent 4 and every prime exponent . Fermat proved FLT for exponent 4. Euler proved FLT for exponent 3[8]. In this paper using automorphic functions we prove FLT for exponents  and , where  is an odd prime. The proof of FLT must be direct. But indirect proof of FLT is disbelieving.

[Chun-Xuan Jiang. Automorphic Functions And Fermat’s Last Theorem（6）. Rep Opinion 2012;4(8):25-27]. (ISSN: 1553-9873). http://www.sciencepub.net/report. 6

doi:10.7537/marsroj040812.06

Keywords: automorphic function; cyclotomic field; Fermat; theorem

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Jiang’s function  in prime distribution

Chun-Xuan Jiang

P. O. Box 3924, Beijing 100854, P. R. China

jcxuan@sina.com

Abstract: We define that prime equations  （5）are polynomials (with integer coefficients) irreducible over integers, where  are all prime. If Jiang’s function  then （5）has finite prime solutions. If  then there are infinitely many primes  such that  are primes. We obtain a unite prime formula in prime distribution   （8）. Jiang’s function is accurate sieve function. Using Jiang’s function we prove about 600 prime theorems [6]. Jiang’s function provides proofs of the prime theorems which are simple enough to understand and accurate enough to be useful.

[Chun-Xuan Jiang. Jiang’s function  in prime distribution. Rep Opinion 2012;4(8):28-34]. (ISSN: 1553-9873). http://www.sciencepub.net/report. 7

doi:10.7537/marsroj040812.07

Keywords: function; Jiang; Prime

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The Hardy-Littlewood prime k-tuple conjecture is false

Chun-Xuan Jiang

P. O. Box 3924, Beijing 100854, P. R. China

jcxuan@sina.com

Abstract: Using Jiang function we prove Jiang prime -tuple theorem. We prove that the Hardy-Littlewood prime -tuple conjecture is false. Jiang prime -tuple theorem can replace the Hardy-Littlewood prime -tuple conjecture.

[Chun-Xuan Jiang. The Hardy-Littlewood prime k-tuple conjecture is false. Rep Opinion 2012;4(8):35-38]. (ISSN: 1553-9873). http://www.sciencepub.net/report. 8

doi:10.7537/marsroj040812.08

Keywords: function; Jiang; prime; theorem; conjecture

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Ecophysiological Responses of Grey Mangrove (Avicennia marina) (Forssk.) Vierh. to Oil Pollution at Ras Mohammed Protective Area

Amal A. Morsy; Amira A. Hassanein and Hawa O. El-Refaai

Botany Department, Faculty of Science, Ain shames University-Egypt

Abstract: The current study was undertaken to investigate the adverse effect of oil spills on the performance of Avicennia marina grow in Ras Mohammed protected Area. The plant materials were collected from four different plots of studied area. The effects of oil pollution on the sediment analysis, photosynthetic pigments, lipid peroxidation as malondialdehyde accumulation, enzyme activities, amino acids, total phenols, total soluble proteins, carbohydrates, and organic acids, of Avicennia marina grow in Ras Mohammed protected Area were analyzed. The data cleared significant decrease in the content chlorophyll a, chlorophyll b, and carotenoids, in Avicennia marina. The Most significant decrease was observed in plot1 and the highest one in plants growing in plot 4. The significant increasing malondialdehyde (MDA) contents in Avicennia marina growing in plot (1) in compassion to Avicennia marina growing in plot (4) sites. The activity of peroxidase was significantly increased under oil pollution in Avicennia marina. The highest value was detected in plot (1) and lowest value was cleared in plot (4) plants. On the other hand, the activity of catalase, superoxide dismutase and glutathione reductase significantly decreased in Avicennia marina growing in plots 1. The highest activity of these enzymes detected in plants growing in plots 4 of the studied area of Ras Mohammed. Avicennia marina growing in plots (3,4) contained higher concentrations of aspartic acid, serine, glutamic acid, glycine, alanine, valine, leucine, lysine, threonine, isoleucine, argnine and ammonia than plots (1,2), while Phenylalanine detected only in plant growing in plots (3,4). Proline content was increased in Avicennia marina plants growing in plots (1, 2) as compared to Avicennia marina plants growing in plots (3, 4). The most significant increases in total phenols, total soluble protein and carbohydrate in Avicennia marina plants growing in plot1 and the significant decreases was observed in plants growing in plots 4. The total carbohydrate contents show a reverse pattern. Oil pollution caused significant increase in the contents of citric acid, jasmonic acid and salicylic acid. The highest value was determined in Avicennia marina plants growing in plot 1 and the lowest value was cleared in plants growing in plot (4). These changes were discussed in relation to plant performance under environmental pollution.

[Amal A. Morsy; Amira A. Hassanein and Hawa O. El-Refaai. Ecophysiological Responses of Grey Mangrove (Avicennia marina) (Forssk.) Vierh. to Oil Pollution at Ras Mohammed Protective Area. Rep Opinion 2012;4(8):43-56]. (ISSN: 1553-9873). http://www.sciencepub.net/report. 10

doi:10.7537/marsroj040812.10

Key words: Avicennia marina, oil spills, Ras Mohammed protected area.

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OVERVIEW OF COPPER TOXICITY TO AQUATIC LIFE

Okocha RO and Adedeji OB

Veterinary Public Health and Preventive Medicine University of Ibadan, Nigeria

roobee27_okocha@yahoo.com, +23428183240

ABSTRACT: Copper is an essential element required by all living organisms, but it can be toxic to aquatic species when present at elevated concentrations. This document is an overview of copper toxicity and its effects on aquatic organisms. The subtopics include history: the copper age; general sources and uses; basic copper chemistry; environmental factors; effects on aquatic organisms; public health implication; proposed criteria for the protection of aquatic life and human health and recommendations.

[Okocha RO and Adedeji OB. OVERVIEW OF COPPER TOXICITY TO AQUATIC LIFE. Rep Opinion 2012;4(8):57-67]. (ISSN: 1553-9873). http://www.sciencepub.net/report. 11

doi:10.7537/marsroj040812.11

Keywords : Aquatic organisms, Copper, Toxicity, Aquatic Poll

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Similarity solution for natural convection from a moving vertical plate with internal heat generation and a convective boundary condition in the presence of thermal radiation and viscous dissipation

Olanrewaju, P.O.

Department of Mathematics, Covenant University, Ota, Nigeria.

oladapo_anu@yahoo.ie, philip.olanrewaju@covenantuniversity.edu.ng

Abstract: In this paper we analyze the influences of thermal radiation, viscous dissipation, buoyancy force and internal heat generation on steady laminar convection flow over a semi-moving vertical plate in the presence of a convective surface boundary condition. In the analysis, we assumed that the left surface of the plate is in contact with a hot fluid while the cold fluid on the right surface of the plate contains a heat source that decays exponentially with the classical similarity variable. We utilized similarity variable to transform the governing non-linear partial differential equations into a system of ordinary differential equations, which are solved numerically by applying shooting iteration technique alongside with sixth order Runge-Kutta integration scheme for better accuracy. The effects of the local Biot number, the Prandtl number, the internal heat generation parameter, the thermal radiation, the local Grashof number and the Eckert number on the velocity and temperature profiles are illustrated and interpreted in physical terms. A comparison with previously published results on similar special cases showed excellent agreement. Finally, numerical values of physical quantities, such as the local skin-friction coefficient and the local Nusselt number are presented in tabular form.

[Olanrewaju, P.O. Similarity solution for natural convection from a moving vertical plate with internal heat generation and a convective boundary condition in the presence of thermal radiation and viscous dissipation. Rep Opinion 2012;4(8):68-76]. (ISSN: 1553-9873). http://www.sciencepub.net/report. 12

doi:10.7537/marsroj040812.12

Key words: : viscous dissipation; thermal radiation; moving vertical plate; local Grashof number; local Biot number; Eckert number; convective surface boundary condition.

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