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:

1. $M = B!$
2. $M = \prod {q^a}$, 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? Note by Nikola Jovanović
1 year, 8 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
• Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

• bulleted
• list

1. numbered
2. list

1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

> This is a quote
This is a quote
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $\boxed{123}$

Sort by:

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.

- 1 year, 7 months ago