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Question Number 22625 by Rasheed.Sindhi last updated on 21/Oct/17

$$\mathrm{For}\mathrm{each}\mathrm{positive}\mathrm{integer}\mathrm{n}\mathrm{define}$$$${\mathrm{a}}_{\mathrm{n}}=30+{\mathrm{n}}^2,\mathrm{and}{\mathrm{d}}_{\mathrm{n}}=\gcd ({\mathrm{a}}_{\mathrm{n}},{\mathrm{a}}_{\mathrm{n}+1}).$$$$\mathrm{Find}\mathrm{the}\mathrm{set}\mathrm{of}\mathrm{all}\mathrm{values}\mathrm{that}\mathrm{are}$$$$\mathrm{taken}\mathrm{by}{\mathrm{d}}_{\mathrm{n}}\mathrm{and}\mathrm{show}\mathrm{by}\mathrm{examples}$$$$\mathrm{that}\mathrm{each}\mathrm{of}\mathrm{these}\mathrm{values}\mathrm{are}\mathrm{attained}.$$

Commented by Tinkutara last updated on 21/Oct/17

$$Probably{d}_n=\{1,11,121\}$$

Commented by Rasheed.Sindhi last updated on 21/Oct/17

$$\mathrm{Are}\mathrm{you}\mathrm{sure}\mathrm{that}{11}^3=1331$$$$\mathrm{or}\mathrm{any}\mathrm{other}\mathrm{number}{\mathrm{doesn}}^{\prime }\mathrm{t}$$$$\mathrm{include}?\mathrm{Or}\mathrm{even}\mathrm{the}\mathrm{above}\mathrm{set}$$$$\mathrm{is}\mathrm{finite}?\mathrm{Can}\mathrm{we}\mathrm{prove}\mathrm{it}?$$$$\mathrm{In}\mathrm{general}\mathrm{if}{\mathrm{a}}_{\mathrm{n}}=\mathrm{k}+{\mathrm{n}}^2,\mathrm{for}\mathrm{what}\mathrm{values}$$$$\mathrm{of}\mathrm{k},{\mathrm{d}}_{\mathrm{n}}\mathrm{is}\mathrm{finite}?$$

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