Solving Problems From The Back - 4

How can we possibly show this? We have such little control over anything. Wait, Does showing that 2Lf(2n)2n2n(modf(2n)) 2^{ L f( 2^n)} \equiv 2^{ n - 2^n} \pmod { f(2^n)} remind us of anything? The Lefthandside makes it so tempting for us to want to apply Fermat's Little Theorem. Oh, dang! If only f(2n)f(2^n) was a prime ...

Such wishful thinking. How do we "make" it a prime? Well, if it isn't a prime, how about we take a prime factor pp. Now, we backtrack our breadcrumbs to fix it. Remember when I said that breadcrumb 2 is too strong?

Breadcrumb 2B: For all nn, for any pf(2n) p \mid f(2^n), then there exists a kk such that pf(k) p \mid f(k) and pf(2k) p \mid f(2^k).

Breadcrumb 3B: Let's classify (possible) candidates for kk. From the 1st well known result, k=2n+Lpk = 2^n + Lp, where LL is any integer, work.

Breadcrumb 4B: We want to show that there is some LL such that pf(22n+Lp) p \mid f(2^{ 2^n + L p}).

Exercise 8: What do you think Breadcrumb 5B looks like?

Ponder this, and then move on to the next note in this set.

Note by Calvin Lin
5 years, 3 months ago

No vote yet
1 vote

  Easy Math Editor

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

[example link](https://brilliant.org)example link
> 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 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...