[tex]\displaystyle\bf\\Datele~problemei:\\n\in N^\star\\n!= produsului tuturor~numerelor~nenule~mai~mici~sau~egale~cu~n\\n!=1\times2\times3\times...\times n\\x=succesorul~lui~n!\\x=n!+1\\\\Se cere:\\\\n=?\\astfel~incat:\\a)~x=numar~prim\\b)~x~\vdots~5\\c)~x=pp~(patrat~perfect)\\\\Rezolvare\!:\\\\a)\\n=1\implies x=1!+1=1+1=2=numar~prim\\\\n=2\implies x=2!+1=2+1=3=numar~prim\\\\n=3\implies x=3!+1=6+1=7=numar~prim\\\\b)\\n=4\implies x=4!+1=24+1=25~\vdots~5[/tex]
.
[tex]\displaystyle\bf\\c)\\n=4\implies x=4!+1=24+1=25=5^2=pp\\\\n=5\implies x=4!+1=120+1=121=11^2=pp[/tex]
