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revised Advice on finding counterexamples
Alternative counterexamples briefly expanded upon in a final comment
Apr
3
comment Advice on finding counterexamples
The only thing I might add is that, at least for me, I try to contradict set size or closure. In this case, you can union any two $2$-element subgroups of the Klein $4$-group, multiply the non-identity elements together, and show that closure fails for the set. (Your answer, quite reasonably, took the set size route instead...)
Apr
2
comment Advice on finding counterexamples
Very nice answer!
Apr
2
awarded  Revival
Apr
1
comment how to show that when an edge is removed from K5, the resluting subgraph is planar.
@Mathemagician1234 Thanks; I expect most would solve it this way (and really it is just Kelvin Soh's comment depicted).
Apr
1
revised how to show that when an edge is removed from K5, the resluting subgraph is planar.
added 281 characters in body
Apr
1
answered how to show that when an edge is removed from K5, the resluting subgraph is planar.
Apr
1
revised Advice on finding counterexamples
deleted 25 characters in body
Apr
1
answered Advice on finding counterexamples
Mar
23
comment When is the number of $N$'s factors $1 + \sqrt{N}$?
Thanks for the pointer! I ended up accepting Robert Israel's argument since it is totally elementary (but also writing up a summary of it that could be presented to non-mathematician school teachers).
Mar
23
comment When is the number of $N$'s factors $1 + \sqrt{N}$?
Thanks for the answer! I've accepted it, but also written up a slightly more detailed version (with some examples) that might be easier for non-mathematicians to digest.
Mar
23
answered When is the number of $N$'s factors $1 + \sqrt{N}$?
Mar
10
revised Prove $(a + b)^2 \geq 4ab$
Eliminating one step for presentation
Mar
9
answered Prove $(a + b)^2 \geq 4ab$
Mar
4
awarded  Enlightened
Mar
4
awarded  Nice Answer
Feb
27
accepted When is the number of $N$'s factors $1 + \sqrt{N}$?
Feb
27
revised When is the number of $N$'s factors $1 + \sqrt{N}$?
tl;dr answer edited in now that R.Israel answered it in the affirmative
Feb
26
comment When is the number of $N$'s factors $1 + \sqrt{N}$?
Do you have a reference offhand? My preference is to have an elementary (and the more accessible, the better) argument; I am concerned that establishing such a bound on $d(N)$ is nontrivial. Moreover, it would still require an additional search. (But, +1 for indicating that the problem will not be intractable...)
Feb
26
comment Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$.
In terms of small edits: I believe that your $P(k)$ is a proposition, so it might not be best to follow it with an equal sign (maybe you could use a colon instead). I found this read slightly strange when you mentioned adding $(k+1)^2$ "to both sides of $P(k)$."