Simple Proof Check - Number Theory

[Fisicks]

Recently I've stumbled upon number theory and decided to order a book which is coming soon so in the mean time I've decided to try what looked like an easy proof to kinda practice on. So tell me where my logic fails or where it surprisingly doesn't. criticism appreciated.

http://img210.imageshack.us/img210/3406/img054i.jpg"

 

[CRGreathouse]

The "investigate" line doesn't justify why
n^2/6 = 6k^2
which is the crux of the proof.

 

[Fisicks]

well i just found a pattern, and it holds.

ex. the 15th natural number squared divisible by 6 is 8100. So K is 15, and you have
8100/6=6*15^2
1350=1350

 

[Fisicks]

Can someone give me more insight on what greathouse said, and how to salvage this proof? Its not like n^2/6=6k^2 isn't true for my conditions.

 

[ramsey2879]

Can someone give me more insight on what greathouse said, and how to salvage this proof? Its not like n^2/6=6k^2 isn't true for my conditions.

try every integer has a unique prime factorization and represent a number divisible by 6 as a product of its prime factors. explain why square numbers have only even exponents in their prime factorization. Use these to justify your logic.