Solving Problems From The Back - 4

How can we possibly show this? We have such little control over anything. Wait, Does showing that $2^{ L f( 2^n)} \equiv 2^{ n - 2^n} \pmod { f(2^n)}$ remind us of anything? The Lefthandside makes it so tempting for us to want to apply Fermat's Little Theorem. Oh, dang! If only $f(2^n)$ 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 $p$ . Now, we backtrack our breadcrumbs to fix it. Remember when I said that breadcrumb 2 is too strong?

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

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

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

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

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

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered 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"

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}`	$\boxed{123}$

Solving Problems From The Back - 4

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