- #1
JoanBraidy
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I proved there's infinitely many n such that S_n has an element of order n^2
Martin Rattigan said:Trivialish.
[itex]3^2+4^2+5^2<3.4.5[/itex].
Suppose [itex]3^{2i}+4^{2i}+5^{2i}<3^i.4^i.5^i[/itex] for [itex]i<n[/itex], then
[itex]3^{2(i+1)}+4^{2(i+1)}+5^{2(i+1)}<25(3^i.4^i.5^i)<3^{i+1}4^{i+1}5^{i+1}[/itex], hence [itex]3^{2n}+4^{2n}+5^{2n}<3^n4^n5^n[/itex] by induction.
It follows that [itex]S_{3^n4^n5^n}[/itex] has an element of order [itex](3^n4^n5^n)^2[/itex] for all [itex]n\in \mathbb{N}[/itex].
Similarly [itex]5^{3n}+7^{3n}+9^{3n}+11^{3n}<5^n7^n9^n11^n[/itex], so there are an infinite number of [itex]k[/itex] such that [itex]S_k[/itex] contains an element of order [itex]k^3[/itex].
I think its probably true that there are an infinite number of [itex]n[/itex] such that [itex]S_n[/itex] contains an element of order [itex]n^k[/itex] for any [itex]k\in \mathbb{N}[/itex].
A trivial result is one that is easily proved or obvious, and does not require any significant effort or insight to understand. It is often seen as uninteresting or unimportant in the context of a scientific study.
A result can be considered trivial if it follows directly from well-known facts, definitions, or assumptions. It may also be considered trivial if it is a simple consequence of basic arithmetic or algebraic operations.
Considering if a result is trivial is important because it allows scientists to determine the significance and relevance of their findings. Trivial results may not add any new knowledge or insights to a field of study, and may not be worth publishing or presenting.
Not necessarily. While trivial results may not be groundbreaking or novel, they can still be useful in certain contexts. For example, a trivial result may serve as a starting point for more complex investigations or can be used to validate more complicated theories.
Yes, it is possible for a result to be both trivial and significant. A result may be considered trivial in one context, but can still have important implications or applications in another context. Additionally, a seemingly trivial result may lead to further research and discoveries that are more significant in the long run.