Question Number 182374 by rs4089 last updated on 08/Dec/22
Commented by Rasheed.Sindhi last updated on 08/Dec/22
$${a}=\mathrm{10},{b}=\mathrm{13},{n}=\mathrm{3197} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{OR} \\ $$$${a}=\mathrm{13},{b}=\mathrm{10},{n}=\mathrm{3197} \\ $$
Answered by Rasheed.Sindhi last updated on 09/Dec/22
![n=a^3 +b^3 =(a+b)(a^2 −ab+b^2 ) ∵ a+b≤a^2 −ab+b^2 [see Q#182379] and 23 is the smallest prime factor of a^3 +b^3 ∴ a+b=23 with a^2 −ab+b^2 has no prime factor less than or equal to 23 Note: By {a_0 ,b_0 } we mean (a_0 ,b_0 ) or (b_0 ,a_0 ). Also we knowt If (a_0 ,b_(0 ) ) is solution so the (b_0 ,a_0 ). Case:{a,b}={1,22} a^2 −ab+b^2 =463=463 ✓ Case:{a,b}={2,21} a^2 −ab+b^2 =403=13_(<23) ×31 × Case:{a,b}={3,20} a^2 −ab+b^2 =349=349 ✓ Case:{a,b}={4,19} a^2 −ab+b^2 =301= 7_(<23) ×43 × Case:{a,b}={5,18} a^2 −ab+b^2 =259=7_(<23) ×37 Case:{a,b}={6,17} a^2 −ab+b^2 =223=223 ✓ Case:{a,b}={7,16} a^2 −ab+b^2 =193=193 ✓ Case:{a,b}={8,15} a^2 −ab+b^2 =169=13_(<23) ×13 Case:{a,b}={9,14} a^2 −ab+b^2 =151=151 ✓ Case:{a,b}={10,13}_(−) a^2 −ab+b^2 =139=139_(−) ✓ Case:{a,b}={11,12} a^2 −ab+b^2 =133=7_(<23) ×19 In underlined case a^2 −ab+b^2 has no prime factor≤23 and also 139 the least number. Hence 23×139=3197 is least value of n and {a,b}={10,13} n=a^3 +b^3 =(a+b)(a^2 −ab+b^2 )=23∙139=3197](Q182420.png)
$${n}={a}^{\mathrm{3}} +{b}^{\mathrm{3}} =\left({a}+{b}\right)\left({a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} \right) \\ $$$$\because\:{a}+{b}\leqslant{a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} \:\left[{see}\:{Q}#\mathrm{182379}\right] \\ $$$$\:\:\:\:{and}\:\mathrm{23}\:{is}\:{the}\:{smallest}\:{prime}\:{factor} \\ $$$$\:\:\:\:\:{of}\:{a}^{\mathrm{3}} +{b}^{\mathrm{3}} \\ $$$$\therefore\:{a}+{b}=\mathrm{23}\:{with}\:{a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} {has}\:{no}\: \\ $$$$\:\:\:\:\:\:{prime}\:{factor}\:{less}\:{than}\:{or}\:{equal}\:{to} \\ $$$$\:\:\:\:\:\:\:\mathrm{23} \\ $$$${Note}:\:{By}\:\left\{{a}_{\mathrm{0}} ,{b}_{\mathrm{0}} \right\}\:{we}\:{mean}\:\left({a}_{\mathrm{0}} ,{b}_{\mathrm{0}} \right)\:{or}\:\left({b}_{\mathrm{0}} ,{a}_{\mathrm{0}} \right). \\ $$$${Also}\:{we}\:{knowt}\:{If}\:\left({a}_{\mathrm{0}} ,{b}_{\mathrm{0}\:} \right)\:{is}\:{solution} \\ $$$${so}\:{the}\:\left({b}_{\mathrm{0}} ,{a}_{\mathrm{0}} \right). \\ $$$$\boldsymbol{\mathrm{Case}}:\left\{{a},{b}\right\}=\left\{\mathrm{1},\mathrm{22}\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} =\mathrm{463}=\mathrm{463}\:\checkmark \\ $$$$\boldsymbol{\mathrm{Case}}:\left\{{a},{b}\right\}=\left\{\mathrm{2},\mathrm{21}\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} =\mathrm{403}=\underset{<\mathrm{23}} {\mathrm{13}}×\mathrm{31}\:× \\ $$$$\boldsymbol{\mathrm{Case}}:\left\{{a},{b}\right\}=\left\{\mathrm{3},\mathrm{20}\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} =\mathrm{349}=\mathrm{349}\:\checkmark \\ $$$$\boldsymbol{\mathrm{Case}}:\left\{{a},{b}\right\}=\left\{\mathrm{4},\mathrm{19}\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} =\mathrm{301}=\:\underset{<\mathrm{23}} {\mathrm{7}}\:×\mathrm{43}\:× \\ $$$$\boldsymbol{\mathrm{Case}}:\left\{{a},{b}\right\}=\left\{\mathrm{5},\mathrm{18}\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} =\mathrm{259}=\underset{<\mathrm{23}} {\mathrm{7}}×\mathrm{37} \\ $$$$\boldsymbol{\mathrm{Case}}:\left\{{a},{b}\right\}=\left\{\mathrm{6},\mathrm{17}\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} =\mathrm{223}=\mathrm{223}\:\checkmark \\ $$$$\boldsymbol{\mathrm{Case}}:\left\{{a},{b}\right\}=\left\{\mathrm{7},\mathrm{16}\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} =\mathrm{193}=\mathrm{193}\:\checkmark \\ $$$$\boldsymbol{\mathrm{Case}}:\left\{{a},{b}\right\}=\left\{\mathrm{8},\mathrm{15}\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} =\mathrm{169}=\underset{<\mathrm{23}} {\mathrm{13}}×\mathrm{13} \\ $$$$\boldsymbol{\mathrm{Case}}:\left\{{a},{b}\right\}=\left\{\mathrm{9},\mathrm{14}\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} =\mathrm{151}=\mathrm{151}\:\checkmark \\ $$$$\underset{−} {\boldsymbol{\mathrm{Case}}:\left\{{a},{b}\right\}=\left\{\mathrm{10},\mathrm{13}\right\}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\underset{−} {\:\:{a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} =\mathrm{139}=\mathrm{139}}\:\checkmark \\ $$$$\boldsymbol{\mathrm{Case}}:\left\{{a},{b}\right\}=\left\{\mathrm{11},\mathrm{12}\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} =\mathrm{133}=\underset{<\mathrm{23}} {\mathrm{7}}×\mathrm{19} \\ $$$$\mathrm{In}\:\mathrm{underlined}\:\mathrm{case}\:{a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} \:{has} \\ $$$${no}\:{prime}\:{factor}\leqslant\mathrm{23}\:{and}\:{also}\:\mathrm{139} \\ $$$${the}\:{least}\:{number}. \\ $$$${Hence}\:\mathrm{23}×\mathrm{139}=\mathrm{3197}\:{is}\:{least}\:{value} \\ $$$${of}\:{n}\:{and}\:\left\{{a},{b}\right\}=\left\{\mathrm{10},\mathrm{13}\right\} \\ $$$${n}={a}^{\mathrm{3}} +{b}^{\mathrm{3}} =\left({a}+{b}\right)\left({a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} \right)=\mathrm{23}\centerdot\mathrm{139}=\mathrm{3197} \\ $$
Answered by Rasheed.Sindhi last updated on 11/Dec/22
$${n}={a}^{\mathrm{3}} +{b}^{\mathrm{3}} ;\:{a},{b}\in\mathbb{N} \\ $$$${Smallest}\:{prime}\:{factor}\:{of}\:{n}\:=\mathrm{23} \\ $$$$\underset{−} {{a},{b},{n}=?\:{if}\:{n}\:{is}\:{the}\:{smllest}.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:} \\ $$$${Let}\:{a}^{\mathrm{3}} +{b}^{\mathrm{3}} =\mathrm{23}{p}\: \\ $$$${where}\:{p}\left(>\mathrm{23}\right)\in\mathbb{P}\:\:{and}\:{the}\:{p}\:{is}\:{the}\:{least} \\ $$$$\:\:\:\:\:\left({a}+{b}\right)\left({a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} \right)=\mathrm{23}{p} \\ $$$$\:\:\:{a}+{b}=\mathrm{23}\:\wedge\:{a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} ={p} \\ $$$${If}\:\left({a}_{\mathrm{0}} ,{b}_{\mathrm{0}} \right)\:{satisfies}\:{then}\:\left({b}_{\mathrm{0}} ,{a}_{\mathrm{0}} \right)\:{also}\:{does}. \\ $$$$\:\:\:\:{b}=\mathrm{23}−{a}\:\wedge\:{a}^{\mathrm{2}} −{a}\left(\mathrm{23}−{a}\right)+\left(\mathrm{23}−{a}\right)^{\mathrm{2}} ={p}>\mathrm{23} \\ $$$$\:\:\:\:\:\:{a}^{\mathrm{2}} −\mathrm{23}{a}+{a}^{\mathrm{2}} +{a}^{\mathrm{2}} −\mathrm{46}{a}+\mathrm{529}>\mathrm{23} \\ $$$$\:\:\mathrm{3}{a}^{\mathrm{2}} −\mathrm{69}{a}+\mathrm{529}>\mathrm{23}\:\wedge\:\mathrm{3}{a}^{\mathrm{2}} −\mathrm{69}{a}+\mathrm{529}\in\mathbb{P} \\ $$$${Let}\:{f}\left({a}\right)=\:\mathrm{3}{a}^{\mathrm{2}} −\mathrm{69}{a}+\mathrm{529} \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{463}\in\mathbb{P}\checkmark \\ $$$${f}\left(\mathrm{2}\right)=\mathrm{403}\notin\mathbb{P} \\ $$$${f}\left(\mathrm{3}\right)=\mathrm{349}\in\mathbb{P}\checkmark \\ $$$${f}\left(\mathrm{4}\right)=\mathrm{301}\notin\mathbb{P} \\ $$$${f}\left(\mathrm{5}\right)=\mathrm{259}\notin\mathbb{P} \\ $$$${f}\left(\mathrm{6}\right)=\mathrm{223}\in\mathbb{P}\checkmark \\ $$$${f}\left(\mathrm{7}\right)=\mathrm{193}\in\mathbb{P}\checkmark \\ $$$${f}\left(\mathrm{8}\right)=\mathrm{169}\notin\mathbb{P} \\ $$$${f}\left(\mathrm{9}\right)=\mathrm{151}\in\mathbb{P}\checkmark \\ $$$${f}\left(\mathrm{10}\right)=\mathrm{139}\in\mathbb{P}\checkmark{the}\:{least} \\ $$$${f}\left(\mathrm{11}\right)=\mathrm{133}\notin\mathbb{P} \\ $$$${a}=\mathrm{10}\Rightarrow{b}=\mathrm{23}−\mathrm{10}=\mathrm{13} \\ $$$${n}={a}^{\mathrm{3}} +{b}^{\mathrm{3}} =\left({a}+{b}\right)\left({a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} \right)=\mathrm{23}.\mathrm{139}=\mathrm{3197} \\ $$