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

For each positive integer n define  a_n =30+n^2 ,and d_n =gcd(a_n ,a_(n+1) ).  Find the set of all values that are  taken by d_n  and show by examples  that each of these values are attained.

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

Commented by Tinkutara last updated on 21/Oct/17

Probably d_n ={1,11,121}

$${Probably}\:{d}_{{n}} =\left\{\mathrm{1},\mathrm{11},\mathrm{121}\right\} \\ $$

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

Are you sure that 11^3 =1331  or any other number doesn′t   include? Or even the above set  is finite? Can we prove it?  In general if a_n =k+n^2 ,for what values  of  k, d_n  is finite?

$$\mathrm{Are}\:\mathrm{you}\:\mathrm{sure}\:\mathrm{that}\:\mathrm{11}^{\mathrm{3}} =\mathrm{1331} \\ $$$$\mathrm{or}\:\mathrm{any}\:\mathrm{other}\:\mathrm{number}\:\mathrm{doesn}'\mathrm{t}\: \\ $$$$\mathrm{include}?\:\mathrm{Or}\:\mathrm{even}\:\mathrm{the}\:\mathrm{above}\:\mathrm{set} \\ $$$$\mathrm{is}\:\mathrm{finite}?\:\mathrm{Can}\:\mathrm{we}\:\mathrm{prove}\:\mathrm{it}? \\ $$$$\mathrm{In}\:\mathrm{general}\:\mathrm{if}\:\mathrm{a}_{\mathrm{n}} =\mathrm{k}+\mathrm{n}^{\mathrm{2}} ,\mathrm{for}\:\mathrm{what}\:\mathrm{values} \\ $$$$\mathrm{of}\:\:\mathrm{k},\:\mathrm{d}_{\mathrm{n}} \:\mathrm{is}\:\mathrm{finite}? \\ $$

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