# Solving Problems From The Back - 5

Breadcrumb 5B: We want to show that there is some $$L$$ such that $$p \mid 2^{ 2^n + Lp} -2^n = 2^n ( 2^{ Lp + 2^n - n } -1)$$.

Breadcrumb 6B: Take $$L = n - 2^n$$. If we can apply Fermat's Little Theorem, then we are done, since $$2^{Lp} \equiv 2^{L} \equiv 2^{n - 2^n} \pmod{p}$$.

Exercise 9: Can we apply Fermat's Little Theorem directly? If no, why not?

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

Note by Calvin Lin
3 years, 10 months ago

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2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
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The text passage contains a mathematical problem in the form of an exercise asking whether Fermat's Little Theorem can be applied directly, but it does not provide an answer to this question.
single_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
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# Solving Problems From The Back - 5

Breadcrumb 5B: We want to show that there is some $$L$$ such that $$p \mid 2^{ 2^n + Lp} -2^n = 2^n ( 2^{ Lp + 2^n - n } -1)$$.

Breadcrumb 6B: Take $$L = n - 2^n$$. If we can apply Fermat's Little Theorem, then we are done, since $$2^{Lp} \equiv 2^{L} \equiv 2^{n - 2^n} \pmod{p}$$.

Exercise 9: Can we apply Fermat's Little Theorem directly? If no, why not?

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

Note by Calvin Lin
3 years, 10 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. 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 1paragraph 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 $$2 \times 3$$
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} $$\boxed{123}$$
user
The text passage contains a mathematical problem in the form of an exercise asking whether Fermat's Little Theorem can be applied directly, but it does not provide an answer to this question.

TYPE: SINGLE_QUESTION