Question Number 173431 by mnjuly1970 last updated on 11/Jul/22
Commented by Rasheed.Sindhi last updated on 11/Jul/22
$$\left({a},{b}\right)=\left(\mathrm{10},\mathrm{13}\right),\left(\mathrm{13},\mathrm{10}\right); \\ $$$${n}=\mathrm{3197}=\mathrm{23}×\mathrm{139} \\ $$
Commented by mr W last updated on 11/Jul/22
$${very}\:{right}\:{sir}! \\ $$
Answered by mr W last updated on 11/Jul/22
![n=a^3 +b^3 =(a+b)[(a+b)^2 −3ab]=A×B A=a+b=23 B=(a+b)^2 −3ab=23^2 −3ab n_(min) ⇒B_(min) ⇒(ab)_(max) (ab)≤(((a+b)/2))^2 =(((23)/2))^2 for a,b ∈R, (ab)_(max) is when a=b=((23)/2). for a,b∈N, (ab)_(max) is when a and b are at most equal to each other, besides B=23^2 −3ab should have no prime factor less than 23. we start with the best a=11, b=12: B=23^2 −3×11×12=133=7×19 ⇒bad! now we try the next best a=10, b=13: B=23^2 −3×10×13=139=prime ⇒ok! bingo! this is a direct hit! ⇒n_(min) =23×139=3197 similary for n_(max) : we start with a=1, b=22: B=23^2 −3×1×22=463=prime⇒ok! ⇒n_(max) =23×463=10649](Q173439.png)
$${n}={a}^{\mathrm{3}} +{b}^{\mathrm{3}} =\left({a}+{b}\right)\left[\left({a}+{b}\right)^{\mathrm{2}} −\mathrm{3}{ab}\right]={A}×{B} \\ $$$${A}={a}+{b}=\mathrm{23} \\ $$$${B}=\left({a}+{b}\right)^{\mathrm{2}} −\mathrm{3}{ab}=\mathrm{23}^{\mathrm{2}} −\mathrm{3}{ab} \\ $$$${n}_{{min}} \Rightarrow{B}_{{min}} \Rightarrow\left({ab}\right)_{{max}} \\ $$$$\left({ab}\right)\leqslant\left(\frac{{a}+{b}}{\mathrm{2}}\right)^{\mathrm{2}} =\left(\frac{\mathrm{23}}{\mathrm{2}}\right)^{\mathrm{2}} \\ $$$${for}\:{a},{b}\:\in{R},\:\left({ab}\right)_{{max}} \:{is}\:{when}\:{a}={b}=\frac{\mathrm{23}}{\mathrm{2}}. \\ $$$${for}\:{a},{b}\in{N},\:\left({ab}\right)_{{max}} \:{is}\:{when}\:{a}\:{and}\:{b}\: \\ $$$${are}\:{at}\:{most}\:{equal}\:{to}\:{each}\:{other},\:{besides} \\ $$$${B}=\mathrm{23}^{\mathrm{2}} −\mathrm{3}{ab}\:{should}\:{have}\:{no}\:{prime}\: \\ $$$${factor}\:{less}\:{than}\:\mathrm{23}.\: \\ $$$${we}\:{start}\:{with}\:{the}\:{best}\:{a}=\mathrm{11},\:{b}=\mathrm{12}:\: \\ $$$${B}=\mathrm{23}^{\mathrm{2}} −\mathrm{3}×\mathrm{11}×\mathrm{12}=\mathrm{133}=\mathrm{7}×\mathrm{19}\:\Rightarrow{bad}! \\ $$$${now}\:{we}\:{try}\:{the}\:{next}\:{best}\:{a}=\mathrm{10},\:{b}=\mathrm{13}: \\ $$$${B}=\mathrm{23}^{\mathrm{2}} −\mathrm{3}×\mathrm{10}×\mathrm{13}=\mathrm{139}={prime}\:\Rightarrow{ok}! \\ $$$${bingo}!\:{this}\:{is}\:{a}\:{direct}\:{hit}! \\ $$$$\Rightarrow{n}_{{min}} =\mathrm{23}×\mathrm{139}=\mathrm{3197} \\ $$$$ \\ $$$${similary}\:{for}\:{n}_{{max}} : \\ $$$${we}\:{start}\:{with}\:{a}=\mathrm{1},\:{b}=\mathrm{22}: \\ $$$${B}=\mathrm{23}^{\mathrm{2}} −\mathrm{3}×\mathrm{1}×\mathrm{22}=\mathrm{463}={prime}\Rightarrow{ok}! \\ $$$$\Rightarrow{n}_{{max}} =\mathrm{23}×\mathrm{463}=\mathrm{10649} \\ $$
Commented by mnjuly1970 last updated on 12/Jul/22
$$\mathrm{very}\:\mathrm{nice}\:\mathrm{solution} \\ $$$$\mathrm{sir}\:\mathrm{W}...\mathrm{grateful}...\mathrm{thanks}\:\mathrm{alot}.. \\ $$
Commented by Rasheed.Sindhi last updated on 12/Jul/22
$$\mathbb{G}\boldsymbol{\mathrm{reat}}\:\boldsymbol{\mathrm{sir}}! \\ $$
Commented by Tawa11 last updated on 13/Jul/22
$$\mathrm{Great}\:\mathrm{sir} \\ $$