# Questions tagged [totient-function]

Questions on the totient function $\phi(n)$ (sometimes $\varphi(n)$) of Euler, the function that counts the number of positive integers relatively prime to and less than or equal to $n$.

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### Can $\phi(n) = \sqrt{\frac{n}{2}}$ for any natural number other than $2$?

I have seen a proof that $\phi(n) \geq \sqrt{\frac{n}{2}}$, and I was wondering if equality can happen occasionally. I used a computer program and I couldn't find any solution up to a million other ...
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### My idea about the number of coprime pairs up to $N$.

Today, I wanted to write a program to count how many integer pairs $(a, b)$ that satisfy: $$1 \leq a < b \leq n, \gcd(a, b) = 1$$ My first instinct was to write a function that check every pair. ...
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This conjecture is inspired by the comment of @Eric Snyder: Prime numbers which end with 03, 23, 43, 63 or 83 $n$ is a natural number $>1$, $\varphi(n)$ denotes the Euler's totient function, $P_n$ ...