Why Euler's Totient function is important?

A few things:

  • You reduce many term polynomials, to just a few terms mod any value. (see here as applied to an integer, a less general form of polynomial).

  • It shreds power towers down to size, with repeated use.

  • It allows us to work in smaller numbers, rather than potentially trillion digit numbers.

  • It allows us to pigeonhole principle coprime variable sets.

  • You can generalize it to products of coprime arithmetic progressions. The product of the first 4 numbers in arithmetic progression 10k+9 have a product that is 1 mod 10 for example (193,401 for those wondering).

  • Can be used to limit long division in finding the reptend length of fractions with coprime denominator in a given base.

  • Cryptography ( forgot this)

    • etc.

Another reason from abstract algebra:

If $n$ is a positive integer then every group of order $n$ is cyclic if and only if $\gcd(n,\varphi(n))=1$.

You can read about the history and proof of this result here.

Tags:

Modular Arithmetic

Elementary Number Theory

Totient Function

Related

If $\int_ {0}^ \infty (\frac{ \sin x}{x})^3 = A$, then $\int_ {0}^ \infty \frac{ x- \sin x}{x^3} = ??$ Integral $\int_0^1 \frac{dx}{\sqrt[3]{x(1-x)}(1-x(1-x))}$ Does a group of order $400$ always have a subgroup of order $200$? Probability Doubt about selection Show that a function satisfies $|f(z)|<1$ for all $z$ in the open left half plane. How to Solve Advanced Recurrence Relations An easy way to solve this limit of a sum? $G$ is a finite group s.t. $(ab)^p=a^pb^p$; $P$ is its $p$-Sylow subgroup. Then there exists $N \unlhd G$ such that $N \cap P = (e)$ and $NP=G$. Groups where no elements commute except for the trivial cases Rationalize the denominator of $\frac{\sqrt{7\sqrt{3}+4\sqrt{5}}}{\sqrt{7\sqrt{3}-4\sqrt{5}}}$ How to factor a fourth degree polynomial Summation containing $\pi$ gives an expression containing $e$

Recent Posts