Proof Contest Day 1

Prove that there no solutions for

$\large{5^a+8^b=c^2 }$

For $a,b,c \in \mathbb{Z^{+}}$ and $b$ is not an even positive integer.

#NumberTheory

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.

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}`	$\boxed{123}$

Comments

$c^2 \equiv 0, 1, 4, 9, 6, 5 \pmod{10}$

$5^a \equiv 5 \pmod{10}$

$8^b \equiv 8, 2 \pmod{10} (b \text{ is odd})$

$5^a + 8^b \equiv 3,7 \pmod{10}$

Clearly, the first statement and the last statement share no residues. Thus proven.

Sharky Kesa - 5 years, 5 months ago

Haha, the same way I solved the question!

Department 8 - 5 years, 5 months ago

Nice problem. By the way , from where have you collected these problems?

Priyanshu Mishra - 5 years, 5 months ago

@Priyanshu Mishra – Own thinking. Try other problems too from this set.

Department 8 - 5 years, 5 months ago

@Department 8 – Trying others.

Priyanshu Mishra - 5 years, 5 months ago

What's mod? And how are 6 and 5 also included?

Yuki Kuriyama - 5 years, 5 months ago

$a \equiv b \pmod c$

Implies a leaves same remainder as b when divided by c.

Department 8 - 5 years, 5 months ago

yes it must have i;e solve by indicies a+b=2

amar nath - 5 years, 5 months ago

The bases are not same. Additionally, there's no multiplication involved so you cannot add the indices.

Arulx Z - 5 years, 4 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}`	$\boxed{123}$