Help: Pollard's p-1, correct way to build M

I need some help regarding one of the steps in Pollard's p-1.

What I understood: We are assuming that there exists $p$ , a prime factor of $n$ , such that $p-1$ is $B$ -powersmooth for some $B$ (not $B$ -smooth!) and that we're trying to build a number $M$ (to be used as an exponent) such that $(p-1)|M$ . Most commonly this is done in two ways:

$M = B!$
$M = \prod {q^a}$ , for all primes $q <= B$ and $a$ sufficiently large.

Let's focus on the second. My confusion comes from the value of a. Using the main assumption we made ( $p-1$ is $B$ -powersmooth) it seems clear that $a = \lfloor log_q{B} \rfloor$ . However, some sources (including Wikipedia) use $a = \lfloor log_q{n} \rfloor$ . Since we can assume $n >> B$ this works too but makes $a$ unnecessarily large.

At first I thought it's just an error on Wikipedia and I made a correction, but I found several revisions in page history that change this both ways ( $B$ to $n$ and back), with no real discussion, which makes me unsure. Can someone explain why $n$ makes more sense than $B$ here, or confirm that $n$ was just an error?

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
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The confusion seems to be whether we're assuming $p-1$ is $B$ -powersmooth or $B$ -smooth, right? If it's $B$ -powersmooth, then we can use the smaller bound, but if it's $B$ -smooth, then we have to use the larger one.

The Wikipedia page cites the smaller bound, but the example ( $n=299,$ $B=5$ ) appears to use the larger bound.

Patrick Corn - 3 years, 3 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}`	$\boxed{123}$