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Question Number 21230 by Tinkutara last updated on 16/Sep/17

$$\mathrm{For}\mathrm{each}\mathrm{positive}\mathrm{integer}n,\mathrm{consider}$$$$\mathrm{the}\mathrm{highest}\mathrm{common}\mathrm{factor}{h}_n\mathrm{of}\mathrm{the}\mathrm{two}$$$$\mathrm{numbers}n!+1\mathrm{and}(n+1)!.\mathrm{For}n<100,$$$$\mathrm{find}\mathrm{the}\mathrm{largest}\mathrm{value}\mathrm{of}{h}_n.$$

Answered by dioph last updated on 17/Sep/17

$$\mathrm{as}\mathrm{the}\mathrm{numbers}\mathrm{from}2\mathrm{until}n\mathrm{do}\mathrm{not}$$$$\mathrm{divide}n!+1,h_n\mathrm{is}\mathrm{either}1\mathrm{or}n+1.$$$$h_n=n+1\iff n!+1=k(n+1)$$$$\iff n!\equiv -1(\mathrm{mod}n+1)$$$$\mathrm{using}\mathrm{Wilsons}\mathrm{theorem},\mathrm{this}\mathrm{happens}$$$$\mathrm{only}\mathrm{when}n+1\mathrm{is}\mathrm{prime}.$$$$\mathrm{The}\mathrm{last}\mathrm{prime}\mathrm{before}100\mathrm{is}97\mathrm{so}$$$$\mathrm{for}96!+1\mathrm{and}97!\mathrm{we}\mathrm{have}{h}_n=97$$

Commented by Tinkutara last updated on 17/Sep/17

$$\mathrm{How}\mathrm{do}\mathrm{you}\mathrm{used}{\mathrm{Wilson}}^{\prime }\mathrm{s}\mathrm{theorem}?$$

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