Number theory

What is the algorithm to find exponent of a prime p in n! ?

#NumberTheory

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

The highest power of p that divides n! is

$\displaystyle\sum_{i=1}^{j}\left \lfloor \frac{n}{p^i} \right \rfloor$

,where j is the biggest integer, for which $p^j<n$

Bogdan Simeonov - 7 years, 4 months ago

I think it should be $p^j \leq n$

Labib Rashid - 7 years, 4 months ago

True. But you could save yourself from the trouble by just putting $\displaystyle \sum_{i=1}^\infty \lfloor\frac{n}{p^i}\rfloor$ .

Mursalin Habib - 7 years, 4 months ago

@Mursalin Habib – He asked for an algorithm. A code with your algo would run for an infinite time.

Labib Rashid - 7 years, 4 months ago

@Labib Rashid – Oh! I think I missed the word 'algorithm'. But I don't think the OP meant algorithm literally [It's tagged with number theory]. Despite that, you're right. That algorithm will go on forever in the natural sense. I just posted that because I think that looks cool :)

Mursalin Habib - 7 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$