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I proved theres infinitely many n such that S_n has an element of order n^2
similar to what I did thanksTrivialish.
[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].