My cryptogram hypothesis

Consider equations, an which ${ A }_{ 1 },{ A }_{ 2 },{ A }_{ 3 },\dots { ,A }_{ n-1 },{ A }_{ n }$ is not necessarily distinct non-zero digits in decimal base and $n, m$ is natural numbers.

1. Prove for $n \ge 5$ that equation below hasn't solution: $\overline { { A }_{ 1 } } +\frac { \overline { { A }_{ 1 }{ A }_{ 2 } } }{ \overline { { A }_{ 2 } } } +\frac { \overline { { A }_{ 1 }{ A }_{ 2 }{ A }_{ 3 } } }{ \overline { { A }_{ 2 }{ A }_{ 3 } } } +\dots +\frac { \overline { { A }_{ 1 }{ A }_{ 2 }{ A }_{ 3 }\dots { A }_{ n-1 }{ A }_{ n } } }{ \overline { { A }_{ 2 }{ A }_{ 3 }\dots { A }_{ n-1 }{ A }_{ n } } } =m$

2. Prove for $n \ge 4$ that equation below hasn't solution: $\overline { { A }_{ 1 } } +\frac { \overline { { A }_{ 1 }{ A }_{ 2 }{ A }_{ 1 } } }{ \overline { { A }_{ 2 } } } +\frac { \overline { { { A }_{ 1 }{ A }_{ 2 }{ A }_{ 3 }{ A }_{ 2 }{ A }_{ 1 } } } }{ \overline { { { A }_{ 2 }{ A }_{ 3 }{ A }_{ 2 } } } } +\dots +\frac { \overline { { { A }_{ 1 }{ A }_{ 2 }\dots { A }_{ n-1 }{ A }_{ n }{ A }_{ n-1 }\dots { A }_{ 2 }{ A }_{ 1 } } } }{ \overline { { A }_{ 2 }\dots { A }_{ n-1 }{ A }_{ n }{ A }_{ n-1 }\dots { A }_{ 2 } } } =m$

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}`	$\boxed{123}$

My cryptogram hypothesis

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