Euler Phi Function/Examples/1,000,000
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Example of Use of Euler $\phi$ Function
- $\map \phi {1 \, 000 \, 000} = 400 \, 000$
where $\phi$ denotes the Euler $\phi$ function.
Proof 1
We have that:
\(\ds 1\,000\,000\) | \(=\) | \(\ds 10^6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 5^6 \times 2^6\) |
From Euler Phi Function of Prime Power:
- $\map \phi {p^n} = p^n \paren {1 - \dfrac 1 p}$
Thus:
\(\ds \map \phi {5^6}\) | \(=\) | \(\ds 5^6 \paren {1 - \dfrac 1 5}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 4 5 5^6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 \times 5^5\) |
and:
\(\ds \map \phi {2^6}\) | \(=\) | \(\ds 2^5\) | Euler Phi Function of Prime Power: Corollary |
From Euler Phi Function is Multiplicative:
\(\ds \map \phi {1 \, 000 \, 000}\) | \(=\) | \(\ds \map \phi {10^6}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi {5^6 \times 2^6}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi {5^6} \map \phi {2^6}\) | Euler Phi Function is Multiplicative | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \times 5^5 \times 2^5\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 \times 10^5\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 400 \, 000\) |
Thus:
- $\map \phi {1 \, 000 \, 000} = 400 \, 000$
Proof 2
We have that:
\(\ds 1\,000\,000\) | \(=\) | \(\ds 10^6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 5^6 \times 2^6\) |
From Euler Phi Function of Integer:
- $\ds \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$
where $p$ denotes a prime number.
So:
\(\ds \map \phi {1\,000\,000}\) | \(=\) | \(\ds 1\,000\,000 \paren {1 - \frac 1 2} \paren {1 - \frac 1 5}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1\,000\,000 \times \frac 1 2 \times \frac 4 5\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 400\,000\) |
Thus:
- $\map \phi {1\,000\,000} = 400\,000$