Question Number 61782 by naka3546 last updated on 08/Jun/19
$${Any}\:\:{integer}\left({s}\right)\:\:{which}\:\:{fulfill} \\ $$$$\:\:\:\:\:\:\:\:\:\:{n}^{\mathrm{5}} \:−\:\mathrm{5}{n}^{\mathrm{3}} \:+\:\mathrm{5}{n}\:+\:\mathrm{1}\:\:\mid\:\:{n}!\:\:\:? \\ $$
Commented by naka3546 last updated on 08/Jun/19
$${please}\:\:{show}\:\:{workings}\:! \\ $$
Commented by MJS last updated on 08/Jun/19
$$\mathrm{again}\:\mathrm{we}\:\mathrm{can}\:\mathrm{only}\:\mathrm{try} \\ $$
Commented by MJS last updated on 08/Jun/19
$${n}=\mathrm{0} \\ $$$$\mathrm{no}\:\mathrm{other}\:\mathrm{solution}\:\mathrm{for}\:{n}\leqslant\mathrm{300} \\ $$
Commented by naka3546 last updated on 08/Jun/19
$${there}\:\:{was}\:\:{n}\:\:{so}\:\:{huge}\:\:{that}\:\:{fulfill}\:\:{it}\:. \\ $$$${But},\:\:{I}\:\:{can}'{t}\:\:{still}\:\:{understand}\:\:{until}\:\:{now}\:. \\ $$$${e}.{g}\::\:\:{Let}\:\:\:{X}_{{m}} \left({n}\right)\:\:=\:\:\left({x}^{{n}} \:+\:\frac{\mathrm{1}}{{x}^{{n}} }\right)^{{m}} \\ $$$${for}\:\:{m}\:=\:\mathrm{5}\:.\:\:\:{Could}\:\:\:{anyone}\:\:{explain}\:\:{to}\:\:{get}\:\:{that}\:\:\:{equation}\:\:{is}\:? \\ $$
Commented by naka3546 last updated on 08/Jun/19
$${or}\:\:{may}\:{be}\:\:{like}\:\:{this} \\ $$$$\:\:\:{X}_{{m}} \left({n}\right)\:\:=\:\:\left({x}^{{m}} \:+\:\frac{\mathrm{1}}{{x}^{{m}} }\right)^{{n}} \\ $$