# Case Study

Hello everybody!

as you know that if , $$a\geq b$$ ,then it is not necessary that $$\phi(a)\geq\phi(b)$$ . Since $$\phi$$ not an increasing function.

But there are some cases in which above inequality holds true.

Case 1

When either of $$a$$ and $$b$$ , suppose lets take $$a$$ as an arbitary prime , then $$b=a+1$$ or $$b=a-1$$. Same follows if $$b$$ is prime.

Case 2

When $$a=b$$

Case 3

When $$a$$ and $$b$$ are both primes and , $$a>b$$

2 years, 3 months ago

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$$Consider\quad f\left( n \right) =\phi (n)\quad \quad \quad Where\quad \phi (n)\quad is\quad Euler\quad Totient\quad Function\\ f\left( 5186 \right) =f\left( 5186+1 \right) =f\left( 5186+2 \right) =2592\quad \\ =>\quad f\left( 5186 \right) =f\left( 5187 \right) =f\left( 5188 \right) =2592\quad \\ 5186\quad is\quad the\quad only\quad number\quad which\quad satisfy\quad f\left( x \right) =f\left( x+1 \right) =f\left( x+2 \right) \\ and\quad is\quad less\quad than\quad { 10 }^{ 10 }.$$

- 2 years, 3 months ago