Eulers Totient Function (Number Theoretic Functions), Right Angle Triangle and Their Applications

Abstract

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

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Copyright

Copyright © 2022 Chirag Gupta. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.