In this paper, we introduce some new theorem and results(section ?,? and ?) on Euler’s Totient Function , Right angle triangle and their applications

Introduction

I. INTRODUCTION(PRELIMINARY)

Then triangle EFG is inverted right angle triangle

II. INEQUALITY RELATION

III. RESULTS RELATED TO HARDY-RAMANUJAN NUMBER, AREA OF RECTANGLE AND TRAPEZIUM AND GOLDEN RATIO.

Hence (b) part is proved.

where T is the area of triangle EFG.

IV. ANGLES OF TRIANGLE EFG IN TERMS OF EULERS PHI FUNCTION

V. OBSERVATIONS

VI. APPLICATION OF EULER’S TOTIENT FUNCTION AND NUMBER THEORETIC FUNCTIONS IN STUDENT’S PENCIL COMPASS

Mathematical instruments are used to understand mathematical constructions and concepts. If we talk about mathematical instruments then there are many mathematical instruments like protractor, ruler, set-square, divider, pencil compass and etc. There are two types of mathematical instruments, one is used by student and other are used by teachers. Mathematical instruments used by teachers are very large as compare to mathematical instruments used by student because they are used in black boards or white boards. The mathematical instruments used by student are smaller because they are usable in their books only. If we talk about the student’s pencil compass, then one can form the circles with a radius of about 1 to 15 centimeters.

Here we have applied the concepts of number theory.

If it is assumed that a circle of radius up to 20 centimeters is being formed by student’s pencil compass, then also the following result will be correct. Normally circles of radius up to 14 or 15 centimeters are formed by the student’s pencil compass.

VII. RESULT

If we draw all the possible circles by student’s pencil compass having radius = r ( r = 1,2,3,4,……,maximum possible radius to be made by student’s pencil compass ) , then 6 is the only number whose every positive proper divisors as a radius follow the relation

φ(2r) + τ(2r) + σ(2r) = greatest integer function of P

and the number 6 (itself) as a radius follow the relation

φ(2r) + τ(2r) + σ(2r) = lowest integer function of P .

φ(2r) + τ(2r) + σ(2r) can also be written as φ(d) + τ(d) + σ(d) ,

where P = π x d , π = 3.1415 (four digits after the decimal point) , P is circumference(perimeter) of circle and d is diameter of circle .

NOTE: (a) Draw all the possible circles by student’s pencil compass means to make circles of each radius which is made by student’s pencil compass .

Where φ(d) is the is the euler’s phi function which denote the number of positive integers not exceeding d that are relatively prime to d .

τ(d) denote the number of positive divisors of d and σ(d) denote the sum of these divisors .

In mathematical form : If r = 1 , φ(2) + τ(2) + σ(2) = greatest integer function of 2 x 3.1415 x 1 = 6 ,

If r = 2 , φ(4) + τ(4) + σ(4) = greatest integer function of 2 x 3.1415 x 2 = 12 ,

If r = 3 , φ(6) + τ(6) + σ(6) = greatest integer function of 2 x 3.1415 x 3 = 18 ,

If r = 6 , φ(12) + τ(12) + σ(12) = lowest integer function of 2 x 3.1415 x 6 = 38 ,

Where 1,2 and 3 are the positive proper divisors of 6 .

VIII. ACKNOWLEDGEMENT

I would like to thanks Dr. Pravin Hudge , Dr. V. Patil , Dr. Vikas Deshmane, and Mandar Khasnis sir for useful comments.

References

[1] Gabor J. Szekely , contests in higher mathematics miklos Schweitzer competitions 1962-1991, page number 244 geometry (Balazs csikos)
[2] Elementary number theory by David M. Burton , TATA MC GRAW – HILL Edition .
[3] Pythagoreans introduction to number theory Ramin Takloo-Bighash , Bartolini springer international publishing , 2018 .
[4] Grade school triangles Jack Scalcut , The American Mathematical Monthly 117(8) , 673-685 , 2010 .
[5] Monthly research problems , 1969-73 Richard K Guy , The American Mathematical Monthly 80(10) , 1120-1128, 1973 .
[6] E.Grosswald , The average order of an arithmetic function , Duke mathematics journal 23(1956) , 41-44 .
[7] To count clean triangles we count on imph(n) , Mizan R Khan , Riaz R Khan arxiv preprint arxiv : 2012.11081 , 2020 .
[8] (2002) Number Theory . In : Mathematics problems and proofs , Springer , Boston , MA.https://doi.org/10.1007/0-306-46963-4_3 .
[9] An Introduction to the Theory of Numbers:G.H. Hardy,Edward M.Wright ,Andrew Wiles , Roger Heathbrown , Joseph Silverman.
[10] An Introduction to the Theory of Numbers Book : Niven , Zuckerman , Montgomery .
[11] A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics ) .
[12] Handbook of mathematical , Scientific and engineering formulas , Tables functions , Graphs , Transforms max fogiel; Research and Education Association , Research and Education Assoc; 1984.
[13] A text book of algebra F.Y.B.Sc. / B.A. (semester – 1) , Sheth Publishers .
[14] An Introduction to the Theory of Numbers Book : Niven , Zuckerman , Montgomery .
[15] Wolfram Web Resources , Eric Weisstein .
[16] Mathematical instrument , WIKIPEDIA ,Gerard L’Estrange Turner Scientific Instruments , 1500-1900:An Introduction (university of California Press , 1998) .
[17] COBB,PAUL.1999. “Individual and Collective Mathematical Development : The case of statistical Data Analysis” . Mathematical Thinking and Learning 1 (1):5-43 .
[18] FENNEMA , ELIZABETH , and ROMBERG , THOMAS , eds.1999. Mathematics Classrooms That Promote Understanding . Mahwah , NJ:Erlbaum.