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if-n-N-gt-2-prove-that-n-1-3-n-2-1-3-3-1-0-mod-8-

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Question Number 154649 by mathdanisur last updated on 20/Sep/21
if n ∈ N^(>2) prove that [((n)^(1/3) + ((n + 2))^(1/3) )^3 ] + 1 = 0 (mod 8)
ifn∈N>2provethat[(n3+n+23)3]+1=0(mod8)
Commented by MJS_new last updated on 20/Sep/21
just a try let n=t−1 ⇒ t≥3 f(t)∈R∧0≤f(t)<1 (((t−1))^(1/3) +((t+1))^(1/3) )^3 +1−f(t)=8t f(t)=1−6t+3((((t−1)^2 (t+1)))^(1/3) +(((t−1)(t+1)^2 ))^(1/3) ) we must show 0≤f(t)<1 ∀ t≥3 f(3)≈.0839 and it′s easy to show that lim_(t→∞) f(t) =1
justatryletn=t−1⇒t⩾3f(t)∈R∧0⩽f(t)<1(t−13+t+13)3+1−f(t)=8tf(t)=1−6t+3((t−1)2(t+1)3+(t−1)(t+1)23)wemustshow0⩽f(t)<1∀t⩾3f(3)≈.0839andit′seasytoshowthatlimt→∞f(t)=1
Commented by mathdanisur last updated on 20/Sep/21
Very nice Ser, thank you
VeryniceSer,thankyou

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