Question Number 61778 by naka3546 last updated on 08/Jun/19
$${Any}\:\:{integer}\left({s}\right)\:\:{which}\:\:{fulfill}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{n}^{\mathrm{3}} \:−\:\mathrm{5}{n}^{\mathrm{2}} \:+\:\mathrm{5}{n}\:+\:\mathrm{1}\:\:\mid\:\:{n}!\:\:\:\:? \\ $$
Answered by MJS last updated on 08/Jun/19
$$\frac{{n}!}{{n}^{\mathrm{3}} −\mathrm{5}{n}^{\mathrm{2}} +\mathrm{5}{n}+\mathrm{1}}\in\mathbb{Z} \\ $$$${n}= \\ $$$$\mathrm{0} \\ $$$$\mathrm{2} \\ $$$$\mathrm{3} \\ $$$$\mathrm{83} \\ $$$$\mathrm{84} \\ $$$$\mathrm{126} \\ $$$$\mathrm{181} \\ $$$$\mathrm{239} \\ $$$$\mathrm{294} \\ $$$$... \\ $$
Commented by naka3546 last updated on 08/Jun/19
$${by}\:\:{trying}\:\:{or}\:\:{by}\:\:{exact}\:\:{solving}\:\:? \\ $$
Commented by MJS last updated on 08/Jun/19
$$\mathrm{we}\:\mathrm{cannot}\:\mathrm{solve}\:\mathrm{exactly},\:\mathrm{we}\:\mathrm{can}\:\mathrm{only}\:\mathrm{try} \\ $$