Euler Totient Function Calculator
Compute Euler totient function phi(n) with prime factorization steps, coprime counts, and RSA cryptography insights.
What Is Euler's Totient Function?
Euler's totient function $\varphi(n)$ counts how many integers from 1 to $n$ are coprime to $n$. Two integers are coprime when their greatest common divisor is 1.
Product Formula
If $n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$, then:
$$\varphi(n) = n \prod_{p \mid n}\left(1 - \frac{1}{p}\right)$$Key Properties
- For prime $p$: $\varphi(p) = p - 1$
- For prime power $p^k$: $\varphi(p^k) = p^k - p^{k-1}$
- If $\gcd(m,n)=1$, then $\varphi(mn)=\varphi(m)\varphi(n)$
- Euler's theorem: $a^{\varphi(n)} \equiv 1 \pmod{n}$ when $\gcd(a,n)=1$
The totient function is central to RSA cryptography because private-key operations rely on $\varphi(n)$ for semiprime moduli. Related tools: use the Prime Factors Calculator and GCF Calculator.
Frequently Asked Questions
What is phi(12)?
$\varphi(12)=4$ because 1, 5, 7, and 11 are the integers between 1 and 12 that share no common factor with 12 other than 1.
Why is phi important in RSA?
RSA key generation uses $\varphi(n)$ for $n=pq$. The private exponent $d$ is chosen so $ed \equiv 1 \pmod{\varphi(n)}$, which enables decryption.
Does exponent matter in the product formula?
No. Only distinct prime factors matter. For example, $\varphi(8)=\varphi(2^3)=8(1-\frac{1}{2})=4$.
What is phi(1)?
By convention, $\varphi(1)=1$ because 1 is coprime to itself.
Related Tools
Random Sentence Generator
Binary Bitwise AND
HTML to Markdown Converter
Statistics Calculator
Percentage Change Word Problems
Parallelogram Calculator
Filter Numbers
Stress Strain Calculator
Cubic Feet Calculator
Frequency Unit Converter
Effective Interest Rate Calculator
Units of Production Depreciation Calculator
JSON Encode Online
Seismic Geophone Calculator
Function Grapher
Annuity Payout Calculator
XML to Go POJO Generator
Split Date Into Intervals
Hexagonal Prism Calculator
Color Desaturation