Question Number 69939 by mr W last updated on 29/Sep/19
$${x},{y},{z}\:\in\:{Z}^{+} \\ $$$${find}\:{all}\:{solutions}\:{of}\:\boldsymbol{{xy}}=\left(\boldsymbol{{x}}+\boldsymbol{{y}}\right)\boldsymbol{{z}} \\ $$
Commented by Rasheed.Sindhi last updated on 29/Sep/19
$${k}\in\mathbb{Z}^{+} \:{in}\:{the}\:{following}. \\ $$$$\left({x},{y},{z}\right)=\left(\mathrm{2}{k},\mathrm{2}{k},{k}\right), \\ $$$$\left(\mathrm{5}{k},\mathrm{20}{k},\mathrm{4}{k}\right),\left(\mathrm{20}{k},\mathrm{5}{k},\mathrm{4}{k}\right), \\ $$$$\left(\mathrm{3}{k},\mathrm{6}{k},\mathrm{2}{k}\right),\left(\mathrm{6}{k},\mathrm{3}{k},\mathrm{2}{k}\right),\left(\mathrm{7}{k},\mathrm{42}{k},\mathrm{6}{k}\right), \\ $$$$\left(\mathrm{4},\mathrm{12},\mathrm{3}\right) \\ $$$$\:^{\bullet\:} {If}\:\left({p},{q},{r}\right)\:{is}\:{a}\:{solution}\:{then}\:\left({q},{p},{r}\right) \\ $$$$\:\:{is}\:{also}\:{solution}. \\ $$$$\:^{\bullet} \:{If}\:\left({p},{q},{r}\right)\:{is}\:{a}\:{solution}\:{then}\:\left({pk},{qk},{rk}\right) \\ $$$$\:\:\:\:{is}\:{also}\:{solution}.{k}\in\mathbb{Z}^{+} \\ $$$$…. \\ $$
Commented by mr W last updated on 29/Sep/19
$${thanks}\:{sir}! \\ $$$${these}\:{are}\:{solutions}.\:{but}\:{how}\:{can}\:{we}\: \\ $$$${express}\:{ALL}\:{possible}\:{solutions}? \\ $$
Commented by Rasheed.Sindhi last updated on 29/Sep/19
$${Sir}\:{at}\:{the}\:{moment}\:{I}\:{have}\:{no}\:{idea}\:{of} \\ $$$${expressing}\:{all}\:{the}\:{possible}\:{solutions}! \\ $$$${However}\:{I}\:{am}\:{thinking}\:{about}\:{it}… \\ $$
Commented by mr W last updated on 29/Sep/19
$${thank}\:{you}\:{for}\:{trying}\:{sir}! \\ $$
Commented by Rasheed.Sindhi last updated on 29/Sep/19
$${Sir}\:{I}\:{think}\:{infinite}\:{solutions}! \\ $$$${I}'{ll}\:{try}\:{to}\:{write}\:{some}\:{observations} \\ $$$${about}\:{such}\:{triplets}\:{which}\:{satisfy} \\ $$$${your}\:{equation}. \\ $$
Commented by mind is power last updated on 29/Sep/19
$${suppose}\:{x}\:{and}\:{y}\:{coprime} \\ $$$$\Rightarrow{x}+{y}\:{and}\:{xy}\:{are}\:{coprim}\: \\ $$$$\Rightarrow{xy}\mid{z}\Rightarrow{z}={xy}\:{and}\:{x}+{y}=\mathrm{1} \\ $$$${now}\:{assume}\:{x}\:{and}\:{y}\:{are}\:{note}\:{coprim} \\ $$$${let}\:{d}={gcd}\left({x},{y}\right)\Rightarrow{x}={dx}',{y}={dy}' \\ $$$${x}',{y}'\:{coprime} \\ $$$$\Rightarrow{d}^{\mathrm{2}} {x}'{y}'={d}\left({x}'+{y}'\right){z} \\ $$$$\Rightarrow{dx}'{y}'=\left({x}'+{y}'\right){z} \\ $$$${x}'{y}'\mid{z}\:\:{cause}\:{x}'+{y}'\:{and}\:{x}'{y}'\:{are}\:{coprime}\: \\ $$$${and}\:{x}'{y}'\mid{dx}'{y}'=\left({x}'+{y}'\right){z} \\ $$$${z}={x}'{y}'.{k} \\ $$$$\Rightarrow{d}={k}\left({x}'+{y}'\right) \\ $$$${let}\:{S}=\left\{{n}\in\mathbb{N}\:\:\:{n}\mid{d}\right\} \\ $$$${for}\:{all}\:{m}\in\mathbb{N}^{\ast} \:{we}\:{can}\:{Whrite}\:{m}={p}+{q}\:{withe}\:{p},{q}\:{coprim}\:{m}=\mathrm{1}+\left({m}−\mathrm{1}\right) \\ $$$${k}={m}\:{withe}\:{m}\in{S} \\ $$$${x}'+{y}'=\frac{{d}}{{m}}\in{S} \\ $$$$\left({x}',{y}'\right)\in\left\{\left({a},{b}\right)\mid{a}+{b}=\frac{{d}}{{m}},{and}\:{gcd}\left({a},{b}\right)=\mathrm{1}\right\} \\ $$$${x}={dx}'={da} \\ $$$${y}={dx}'={db} \\ $$$${z}={kx}'{y}'={mab} \\ $$$$ \\ $$
Commented by Prithwish sen last updated on 30/Sep/19
$$ \\ $$$$\mathrm{Given} \\ $$$$\mathrm{xy}=\mathrm{xz}+\mathrm{yz} \\ $$$$\Rightarrow\:\mathrm{2xy}−\mathrm{2xz}−\mathrm{2yz}\:=\:\mathrm{0}\:……\left(\mathrm{i}\right) \\ $$$$\mathrm{and}\:\mathrm{2xy}+\mathrm{2xz}+\mathrm{2yz}=\left(\mathrm{x}+\mathrm{y}+\mathrm{z}\right)^{\mathrm{2}} −\left(\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} \right)…..\left(\mathrm{ii}\right) \\ $$$$\mathrm{Adding}\:\left(\mathrm{i}\right)\:\mathrm{and}\:\left(\mathrm{ii}\right) \\ $$$$\Rightarrow\mathrm{xy}=\frac{\left(\mathrm{x}+\mathrm{y}+\mathrm{z}\right)^{\mathrm{2}} −\left(\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} \right)}{\mathrm{4}}\:…..\left(\mathrm{iii}\right) \\ $$$$\boldsymbol{\mathrm{Now}}\:\boldsymbol{\mathrm{there}}\:\boldsymbol{\mathrm{are}}\:\boldsymbol{\mathrm{three}}\:\boldsymbol{\mathrm{possibilities}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{x}}\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{y}}\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{z}} \\ $$$$\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{I}}\:\:\:\:\:\boldsymbol{\mathrm{even}}\:\:\:\:\:\:\boldsymbol{\mathrm{odd}}\:\:\:\:\:\boldsymbol{\mathrm{even}} \\ $$$$\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{I}}\mathrm{I}\:\:\:\boldsymbol{\mathrm{even}}\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{even}}\:\:\:\boldsymbol{\mathrm{odd}} \\ $$$$\boldsymbol{\mathrm{CaseIII}}\:\:\:\boldsymbol{\mathrm{even}}\:\:\:\:\:\boldsymbol{\mathrm{even}}\:\:\:\:\boldsymbol{\mathrm{even}} \\ $$$$\boldsymbol{\mathrm{Now}}\:\boldsymbol{\mathrm{here}}\:\boldsymbol{\mathrm{I}}\:\boldsymbol{\mathrm{am}}\:\boldsymbol{\mathrm{considering}}\:\boldsymbol{\mathrm{only}}\:\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{I}}\: \\ $$$$\mathrm{Let}\:\mathrm{x}=\:\mathrm{2m}\:\:\mathrm{y}=\:\mathrm{2n}+\mathrm{1}\:\:\mathrm{z}\:=\:\mathrm{2p} \\ $$$$\:\:\:\:\:\Rightarrow\:\:\mathrm{p}\:=\:\frac{\mathrm{m}\left(\mathrm{2n}+\mathrm{1}\right)}{\mathrm{2}\left(\mathrm{m}+\mathrm{n}\right)+\mathrm{1}} \\ $$$$\Rightarrow\:\mathrm{2}\boldsymbol{\mathrm{mp}}+\mathrm{2}\boldsymbol{\mathrm{np}}+\boldsymbol{\mathrm{p}}\:=\:\mathrm{2}\boldsymbol{\mathrm{mn}}\:+\boldsymbol{\mathrm{m}}\:……..\left(\boldsymbol{\mathrm{iv}}\right) \\ $$$$\mathrm{Now}\:\mathrm{from}\:\left(\mathrm{iii}\right) \\ $$$$\:\:\boldsymbol{\mathrm{xy}}\:=\:\frac{\left(\mathrm{2}\boldsymbol{\mathrm{m}}+\mathrm{2}\boldsymbol{\mathrm{n}}+\mathrm{2}\boldsymbol{\mathrm{p}}+\mathrm{1}\right)^{\mathrm{2}} −\left\{\mathrm{4}\boldsymbol{\mathrm{m}}^{\mathrm{2}} +\left(\mathrm{2}\boldsymbol{\mathrm{n}}+\mathrm{1}\right)^{\mathrm{2}} +\mathrm{4}\boldsymbol{\mathrm{p}}^{\mathrm{2}} \right\}}{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{8}\boldsymbol{\mathrm{mn}}+\mathrm{8}\boldsymbol{\mathrm{mp}}+\mathrm{8}\boldsymbol{\mathrm{np}}+\mathrm{4}\boldsymbol{\mathrm{m}}+\mathrm{4}\boldsymbol{\mathrm{p}}}{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\left(\:\mathrm{2}\boldsymbol{\mathrm{mp}}+\mathrm{2}\boldsymbol{\mathrm{np}}+\boldsymbol{\mathrm{p}}\right)+\mathrm{2}\boldsymbol{\mathrm{mn}}+\boldsymbol{\mathrm{m}}\:=\:\mathrm{4}\boldsymbol{\mathrm{mn}}\:+\mathrm{2}\boldsymbol{\mathrm{m}} \\ $$$$\:\:\:\mathrm{Now}\:\because\:\boldsymbol{\mathrm{p}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{an}}\:\boldsymbol{\mathrm{integer}} \\ $$$$\therefore\:\mathrm{p}\:=\:\mathrm{n}+\:\frac{\mathrm{m}−\mathrm{2n}^{\mathrm{2}} −\mathrm{n}}{\mathrm{2m}+\mathrm{2n}+\mathrm{1}} \\ $$$$\Rightarrow\mathrm{m}−\mathrm{2n}^{\mathrm{2}} −\mathrm{n}=\mathrm{0} \\ $$$$\Rightarrow\:\boldsymbol{\mathrm{n}}\:=\:\frac{\sqrt{\mathrm{1}+\mathrm{8}\boldsymbol{\mathrm{m}}}−\mathrm{1}}{\mathrm{4}}\:\:=\:\frac{\left(\mathrm{1}+\mathrm{4}\boldsymbol{\mathrm{q}}\right)−\mathrm{1}}{\mathrm{4}}\:=\:\boldsymbol{\mathrm{q}}\:\:\boldsymbol{\mathrm{putting}}\:\:\boldsymbol{\mathrm{m}}\:=\:\mathrm{2}\boldsymbol{\mathrm{q}}^{\mathrm{2}} +\boldsymbol{\mathrm{q}} \\ $$$$\mathrm{p}\:=\frac{\left(\mathrm{2}\boldsymbol{\mathrm{q}}^{\mathrm{2}} +\boldsymbol{\mathrm{q}}\right)\left(\mathrm{2}\boldsymbol{\mathrm{q}}+\mathrm{1}\right)}{\mathrm{2}\left\{\left(\mathrm{2}\boldsymbol{\mathrm{q}}^{\mathrm{2}} +\boldsymbol{\mathrm{q}}\right)+\boldsymbol{\mathrm{q}}\right\}+\mathrm{1}}\:=\:\frac{\boldsymbol{\mathrm{q}}\left(\mathrm{2}\boldsymbol{\mathrm{q}}+\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{2}\boldsymbol{\mathrm{q}}\left(\mathrm{2}\boldsymbol{\mathrm{q}}+\mathrm{1}\right)+\mathrm{2}\boldsymbol{\mathrm{q}}+\mathrm{1}} \\ $$$$\:\:=\boldsymbol{\mathrm{q}} \\ $$$$\therefore\:\boldsymbol{\mathrm{x}}\:=\:\mathrm{2}\boldsymbol{\mathrm{q}}\left(\mathrm{2}\boldsymbol{\mathrm{q}}+\mathrm{1}\right)\:,\:\boldsymbol{\mathrm{y}}\:=\mathrm{2}\boldsymbol{\mathrm{q}}+\mathrm{1}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{z}}\:=\:\mathrm{2}\boldsymbol{\mathrm{q}} \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\mathrm{q}}\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{x}}\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{y}}\:\:\:\:\:\:\:\boldsymbol{\mathrm{z}} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{6}\:\:\:\:\:\:\:\:\:\:\mathrm{3}\:\:\:\:\:\:\:\mathrm{2} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{2}\:\:\:\:\:\:\mathrm{20}\:\:\:\:\:\:\:\:\:\mathrm{5}\:\:\:\:\:\:\:\:\mathrm{4} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{3}\:\:\:\:\:\:\mathrm{42}\:\:\:\:\:\:\:\:\:\mathrm{7}\:\:\:\:\:\:\:\:\mathrm{6} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{4}\:\:\:\:\:\:\mathrm{72}\:\:\:\:\:\:\:\:\mathrm{9}\:\:\:\:\:\:\:\mathrm{8} \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{so}}\:\boldsymbol{\mathrm{on}}\:……… \\ $$$$\boldsymbol{\mathrm{m}},\boldsymbol{\mathrm{n}},\boldsymbol{\mathrm{p}}\:,\boldsymbol{\mathrm{q}}\:\boldsymbol{\mathrm{are}}\:\boldsymbol{\mathrm{integers}} \\ $$$$\boldsymbol{\mathrm{Mr}}.\:\boldsymbol{\mathrm{W}}\:\boldsymbol{\mathrm{Sir}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{Rasheed}}\:\boldsymbol{\mathrm{Sir}}\:\boldsymbol{\mathrm{please}}\:\boldsymbol{\mathrm{give}}\:\boldsymbol{\mathrm{your}} \\ $$$$\boldsymbol{\mathrm{valuable}}\:\boldsymbol{\mathrm{suggestions}}.\boldsymbol{\mathrm{Sir}}\:\boldsymbol{\mathrm{I}}\:\:\boldsymbol{\mathrm{make}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{whole}}\:\boldsymbol{\mathrm{thing}}\: \\ $$$$\boldsymbol{\mathrm{only}}\:\boldsymbol{\mathrm{dependent}}\:\boldsymbol{\mathrm{on}}\:\boldsymbol{\mathrm{q}}.\:\boldsymbol{\mathrm{By}}\:\boldsymbol{\mathrm{putting}}\:\boldsymbol{\mathrm{q}}=\mathrm{1},\mathrm{2},… \\ $$$$\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{you}}\:\boldsymbol{\mathrm{get}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{result}}. \\ $$
Commented by mr W last updated on 29/Sep/19
$${sir}\:{mind}\:{is}\:{power}: \\ $$$${thanks}\:{for}\:{attempting}\:{sir}! \\ $$$${but} \\ $$$${x}={da} \\ $$$${y}={db} \\ $$$${z}={kd}^{\mathrm{2}} {ab} \\ $$$${don}'{t}\:{satisfy}\:{xy}=\left({x}+{y}\right){z}.\:{besides} \\ $$$${how}\:{to}\:{get}\:{a},{b},{d},{k}? \\ $$
Commented by mind is power last updated on 29/Sep/19
$${i}\:{fix}\:{i}\:{switch}\:{z}\:{and}\:{k}\: \\ $$
Commented by mr W last updated on 29/Sep/19
$${prithwish}\:{sir}: \\ $$$${thanks}\:{for}\:{trying}! \\ $$$${with}\:{x}=\mathrm{2}{m},\:{y}=\frac{\sqrt{\mathrm{1}+\mathrm{8}{m}}+\mathrm{1}}{\mathrm{2}},{z}=\mathrm{2}{p} \\ $$$${if}\:{you}\:{take}\:{m}=\mathrm{2},\:{y}\neq{integer}! \\ $$
Commented by mr W last updated on 29/Sep/19
$${there}\:{are}\:{three}\:{unknowns}\:{x},{y},{z}. \\ $$$${a}\:{general}\:{solution}\:{must}\:{have}\:{two} \\ $$$${parameters}\:{in}\:{the}\:{expression}.\:{this} \\ $$$${is}\:{like}\:{to}\:{find}\:{the}\:{general}\:{integer} \\ $$$${solution}\:{for}\:{x}+\mathrm{2}{y}+{z}=\mathrm{7}. \\ $$
Commented by mind is power last updated on 29/Sep/19
$${d}\:{given}\:{parameter} \\ $$$${z}={dab} \\ $$$${m}\:{divusor}\:{of}\:{d}\: \\ $$$${a}\:{and}\:{b}\:{are}\:{such}\:{a}+{b}=\frac{{d}}{{m}} \\ $$$${and}\:{gcd}\left({a},{b}\right)=\mathrm{1} \\ $$$${a}=\frac{{d}}{{m}}−{b} \\ $$$${gcd}\left({a},{b}\right)=\mathrm{1}\Rightarrow{gcd}\left(\frac{{d}}{{m}},{b}\right) \\ $$$${soit}\:{b}\:{is}\:{coprime}\:{withe}\:\frac{{d}}{{m}\:}\:{withe}\:{b}\leqslant\frac{{d}}{{m}} \\ $$$${same}\:{withe}\:{a}\:{coprime}\:{withe}\:\frac{{d}}{{m}} \\ $$$$\left({a},{b}\right)\ast{describe}\:{all}\:{integer}\:{withe}\:{a}+{b}=\frac{{d}}{{m}}\:{withe}\:{a}\:{and}\:{b}\:{coprime}\:{withe}\:\frac{{d}}{{m}} \\ $$$${exampl}\:{a}+{b}=\mathrm{6} \\ $$$$\left({a},{b}\right)\in\left\{\left(\mathrm{1},\mathrm{5}\right)\:\left(\mathrm{5},\mathrm{1}\right)\right\} \\ $$$${Sorry}\:{my}\:{english}\:{not}\:{Good} \\ $$$$ \\ $$$$ \\ $$
Answered by Rasheed.Sindhi last updated on 02/Oct/19
$${I}\:{have}\:{made}\:{a}\:{formula}\:{which}\:{produce} \\ $$$${a}\:{triplet}\:{satisfying}\:{the}\:{given}\:{equation} \\ $$$${for}\:{every}\:{pair}\:{of}\:{positive}\:{integers}: \\ $$$$\:{Let}\:{p},{q}\in\mathbb{Z}^{+} \\ $$$${x}={p}\left({p}+{q}\right) \\ $$$${y}={q}\left({p}+{q}\right) \\ $$$${z}={pq} \\ $$$${We}\:{can}\:{algebraically}\:{prove}\:{that} \\ $$$$\left({x},{y},{z}\right)=\left(\:{p}\left({p}+{q}\right)\:,\:{q}\left({p}+{q}\right)\:,\:{pq}\:\right)\:{is}\:{solution} \\ $$$${of}\:\:\boldsymbol{{xy}}=\left(\boldsymbol{{x}}+\boldsymbol{{y}}\right)\boldsymbol{{z}} \\ $$$${This}\:{proves}\:{that}\:{the}\:{solutions}\:{are} \\ $$$${infinite}. \\ $$$$\boldsymbol{{Claim}}: \\ $$$$\:^{\bullet} {If}\:\:{p}\:\&\:{q}\:{be}\:{coprime}\:{then} \\ $$$$\left({x},{y},{z}\right)\:{will}\:{be}\:{primitive}\:{triple}\:{satisfying} \\ $$$${xy}=\left({x}+{y}\right){z} \\ $$$$\:^{\bullet} {The}\:{above}\:{formulas}\:{can}\:{produce} \\ $$$$\:\:{all}\:{the}\:{possible}\:{primitive}\:{solutions} \\ $$$$\:^{\bullet} {Non}-{primitive}\:{solution}\:{are}\:{merely} \\ $$$$\:\:{integral}\:{multiples}\:{of}\:{primitive}\:{solutions}. \\ $$$$ \\ $$$$\:\:\:{Your}\:{offered}\:{example}\left(\:\left(\mathrm{8},\mathrm{24},\mathrm{6}\right)\right)\:{is} \\ $$$$\:\:\:\:{is}\:{non}-{primitive}\:{solution}\:{and}\:{its} \\ $$$$\:\:\:\:{corresponding}\:{primitive}\:{solution} \\ $$$$\:\:\:{is}\:\left(\mathrm{4},\mathrm{12},\mathrm{3}\right)\:{and}\:{this}\:{can}\:{be}\:{produced} \\ $$$$\:\:\:{by}\:{my}\:{formulas}\:\left({put}\:{p}=\mathrm{1},{q}=\mathrm{3}\right). \\ $$$${How}\:{do}\:{you}\:{see}\:{this}\:{claim}.{please} \\ $$$${give}\:{a}\:{counter}\:{example}\:{if}\:{you}\:{don}'{t} \\ $$$${agree}. \\ $$
Commented by Prithwish sen last updated on 30/Sep/19
$$\mathrm{As}\:\mathrm{a}\:\mathrm{whole} \\ $$$$\mathrm{for}\:\mathrm{the}\:\mathrm{expression}\: \\ $$$$\boldsymbol{\mathrm{xy}}=\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right)\boldsymbol{\mathrm{z}}\:\:\:\:\boldsymbol{\mathrm{x}},\boldsymbol{\mathrm{y}},\boldsymbol{\mathrm{z}}\:\in\mathbb{Z}^{+} \\ $$$$\mathrm{we}\:\mathrm{have} \\ $$$$\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{I}}\:\boldsymbol{\mathrm{x}}=\mathrm{2}\boldsymbol{\mathrm{q}}\left(\mathrm{2}\boldsymbol{\mathrm{q}}+\mathrm{1}\right)\:\:\boldsymbol{\mathrm{y}}\:=\:\left(\mathrm{2}\boldsymbol{\mathrm{q}}+\mathrm{1}\right)\:\:\boldsymbol{\mathrm{z}}=\mathrm{2}\boldsymbol{\mathrm{q}} \\ $$$$\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{II}}\:\boldsymbol{\mathrm{x}}\:=\:\mathrm{2}\boldsymbol{\mathrm{q}}\:\:\:\:\:\boldsymbol{\mathrm{y}}\:=\:\mathrm{2}\boldsymbol{\mathrm{q}}\left(\mathrm{2}\boldsymbol{\mathrm{q}}−\mathrm{1}\right)\:\:\boldsymbol{\mathrm{z}}\:=\:\mathrm{2}\boldsymbol{\mathrm{q}}−\mathrm{1} \\ $$$$\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{III}}\:\boldsymbol{\mathrm{no}}\:\boldsymbol{\mathrm{such}}\:\boldsymbol{\mathrm{expression}}\:\boldsymbol{\mathrm{found}}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{x}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{y}}\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{z}} \\ $$$$\boldsymbol{\mathrm{q}}=\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\:\:\:\:−\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{I}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}−\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{II}} \\ $$$$\boldsymbol{\mathrm{q}}=\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{20}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{5}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{4}\:\:−\:\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{I}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{4}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{12}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}\:−\:\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{II}}\:\:\:\left(\boldsymbol{\mathrm{value}}\:\boldsymbol{\mathrm{of}}\right. \\ $$$$\boldsymbol{\mathrm{q}}=\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{42}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{7}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{6}−\:\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{I}}\:\:\:\:\:\:\:\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{y}}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{30}\:\:\:\:\:\:\:\:\:\:\:\mathrm{5}\:−\:\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{II}}\:\:\:\:\boldsymbol{\mathrm{can}}\:\boldsymbol{\mathrm{be}}\: \\ $$$$\left.\boldsymbol{\mathrm{q}}=\mathrm{4}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{72}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{9}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{8}\:−\:\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{I}}\:\:\:\:\:\boldsymbol{\mathrm{altered}}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{8}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{56}\:\:\:\:\:\:\:\:\:\:\:\mathrm{7}\:−\:\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{II}} \\ $$$$\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{so}}\:\boldsymbol{\mathrm{on}}…… \\ $$$$\boldsymbol{\mathrm{It}}\:\boldsymbol{\mathrm{confirms}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{expression}}\:\boldsymbol{\mathrm{has}}\:\boldsymbol{\mathrm{infinite}} \\ $$$$\boldsymbol{\mathrm{number}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{solutions}}\:. \\ $$
Commented by Rasheed.Sindhi last updated on 03/Oct/19
$${Sir}\:{how}\:{can}\:{we}\:{prove}\:{that}\:{the}\:{above} \\ $$$${formulas}\:{can}\:{produce}\:\boldsymbol{{all}}\:{possible} \\ $$$${solutions}? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\& \\ $$$${What}'{s}\:{the}\:{sourse}\:{of}\:{problem}? \\ $$
Commented by mr W last updated on 29/Sep/19
$${that}'{s}\:{a}\:{nice}\:{way}\:{sir}! \\ $$$${from}\:{given}\:{p}\:{and}\:{q}\:{we}\:{can}\:{construct} \\ $$$${a}\:{solution}.\:{but}\:{there}\:{could}\:{be}\:{more} \\ $$$${than}\:{just}\:{one}\:{solution}\:{for}\:{given}\:{p},\:{q}. \\ $$$${e}.{g}.\:{z}=\mathrm{6} \\ $$$$\left({x},{y}\right)=\left(\mathrm{10},\mathrm{15}\right)\:{is}\:{a}\:{solution},{but} \\ $$$$\left(\mathrm{8},\mathrm{24}\right)\:{is}\:{also}\:{a}\:{solution}. \\ $$
Commented by Rasheed.Sindhi last updated on 30/Sep/19