iosr-Jm   Volume-7 ~ Issue-4

Abstract: Here an attempt is made to find positive numbers a,b and c such that an+bn=cn. Two cases were considered. an+bn is not divisible by (a+b)2 and an+bn is divisible by (a+b)2. From this a solution for the case n=3 is obtained as (9+5)3 + (9-5)3 = 123. A condition for finding such numbers for any n is also reached.

Key words: Fermat's Last Theorem, Number Theory, Mathematics, Diophantine equation, Pythagorean triples

[1]. Stewart, I. and Tall, D. Algebraic Number Theory . (Chapman & Hall, 1994)
[2]. www-gap.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html. Accessed on 02-08-2004. [online].

Abstract: Writing a standard scientific researchable paper be it project, technical report, dissertation or journal that reveal the application of science and technology to millennium challenges seems to be an herculean task among some scholars and students. Based on the above assertion, this paper critically examined the core aspect of a standard paper that is, methodology. Methodology is the framework and live-wire of any scientific paper. Hence, this paper takes a step to x- ray the components of methodology such as research design, study population, operational variables, sampling technique, and sample size determination, method of date collection and method of data processing/analysis with a view to harnessing the components together in their application for science and technology to millennium challenges.

Key words: research design, study population, sample size, data collection, variable(s) and methodology.

[1]. Creswell, J.W (2003). Research Design: Sage publications, London
[2]. Miles, M.B et al (1974). Quantitative Data Analysis 2nd edition: Sage publications, London
[3]. Altman, D.G (1991). Practical Statistics for Medical Research: Chapman and Hall, London.
[4]. Colton, T. (1974). Statistics in Medicine: Boston Little, Brown and Company 9INC.
[5]. Corlien, M.V; Indra Pathmanathan and Ann Brownlee (2003). Designing and conducting health systems research projects: Volume 1 Proposal development and fieldwork: KIT/IDRC
[6]. Degu, G and Tessema, F. Biostatistics for Health Science Students: lecture note series: Department of Community Health, Jimma Institute of Health Sciences.
[7]. Development in primary health care (1998). Ethical considerations in research focus ESTC-EPHA/CDC PROJECT (2004), Training modules on health research.
[8]. Fletcher, M (1992), Principles and Practice of Epidemiology. Department of Community Health, Faculty of medicine, Addis Ababa University.
[9]. Manktelow, B; Hewitt, M.J and Spiers, N (2002). An introduction to Practical Statistics Using SPSS: Trent Focus, Gondar (1995). Manual for field training Jimma (1996). Manual for student research project
[10]. Mathers, Nigel; Howe, Amanda and Hunn, Amanda. Trent focus for research and The Carter Center 9EPHTI), Addis Ababa.

Abstract: The main thrust of this paper is to study the biquadratic equation with four unknowns ( ) 1 2 x  y  z  w  xyzw . We present six different infinite families of positive integral solutions to this equation.

[1] L.E.Dickson, History of Theory of Numbers, Chelsea Publishing company, New York, Vol.11, (1952).

[2] L.J.Mordell, Diophantine equations, Academic Press, London(1969).

[3] Andreescu, T.Andrica, D., An Intrduction to Diophantine Equations, GIL Publishing house, 2002.

[4] Andreescu ,T., A note on the equation x  y  z  xyz 2 ( ) , General Mathematics Vol.10, No.3-4 ,17-22,(2002).

[5] M.A.Gopalan, S.Vidhyalakshmi, A.Kavitha, "observations on x  y  z  xyz 2 ( ) , accepted in IJMSA.

Abstract: The condition for which ) = 0 (2 ) 1 ) ( (2 1) 1 ( 2 =1 2 n=1 n Z n n Z      where Z is a complex number reveals those points Z for which the functions ) (2 ) 1 ) ( (2 1) 1 ( n=1 Z n=1 n Z i n      and ) (2 ) 1 ) ( (2 1) 1 ( n=1 Z n=1 n Z i n      have zeroes.Finally,by direct analysis we can find zeroes of Riemann zeta funtion .

[1] KONRAD KNOPP.THEORY OF FUNCTIONS,5TH Edition.NEW YORK.DOVER PUBLICATIONS.
[2] JOHN B.CONWAY.FUNCTIONS OF ONE COMPLEX VARIABLE,2nd Edition.NAROSA PUBLISHING HOUSE PVT LTD.
[3] J.N.SHARMA.FUNCTIONS OF A COMPLEX VARIABLE,4th Edition.Krishna Prakashan Media(P) Ltd.