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Question Number 69939 by mr W last updated on 29/Sep/19

x,y,z ∈ Z^+   find all solutions of xy=(x+y)z

$${x},{y},{z}\:\in\:{Z}^{+} \\ $$$${find}\:{all}\:{solutions}\:{of}\:\boldsymbol{{xy}}=\left(\boldsymbol{{x}}+\boldsymbol{{y}}\right)\boldsymbol{{z}} \\ $$

Commented by Rasheed.Sindhi last updated on 29/Sep/19

k∈Z^+  in the following.  (x,y,z)=(2k,2k,k),  (5k,20k,4k),(20k,5k,4k),  (3k,6k,2k),(6k,3k,2k),(7k,42k,6k),  (4,12,3)  ^(• ) If (p,q,r) is a solution then (q,p,r)    is also solution.  ^•  If (p,q,r) is a solution then (pk,qk,rk)      is also solution.k∈Z^+   ....

$${k}\in\mathbb{Z}^{+} \:{in}\:{the}\:{following}. \\ $$$$\left({x},{y},{z}\right)=\left(\mathrm{2}{k},\mathrm{2}{k},{k}\right), \\ $$$$\left(\mathrm{5}{k},\mathrm{20}{k},\mathrm{4}{k}\right),\left(\mathrm{20}{k},\mathrm{5}{k},\mathrm{4}{k}\right), \\ $$$$\left(\mathrm{3}{k},\mathrm{6}{k},\mathrm{2}{k}\right),\left(\mathrm{6}{k},\mathrm{3}{k},\mathrm{2}{k}\right),\left(\mathrm{7}{k},\mathrm{42}{k},\mathrm{6}{k}\right), \\ $$$$\left(\mathrm{4},\mathrm{12},\mathrm{3}\right) \\ $$$$\:^{\bullet\:} {If}\:\left({p},{q},{r}\right)\:{is}\:{a}\:{solution}\:{then}\:\left({q},{p},{r}\right) \\ $$$$\:\:{is}\:{also}\:{solution}. \\ $$$$\:^{\bullet} \:{If}\:\left({p},{q},{r}\right)\:{is}\:{a}\:{solution}\:{then}\:\left({pk},{qk},{rk}\right) \\ $$$$\:\:\:\:{is}\:{also}\:{solution}.{k}\in\mathbb{Z}^{+} \\ $$$$.... \\ $$

Commented by mr W last updated on 29/Sep/19

thanks sir!  these are solutions. but how can we   express ALL possible solutions?

$${thanks}\:{sir}! \\ $$$${these}\:{are}\:{solutions}.\:{but}\:{how}\:{can}\:{we}\: \\ $$$${express}\:{ALL}\:{possible}\:{solutions}? \\ $$

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...

$${Sir}\:{at}\:{the}\:{moment}\:{I}\:{have}\:{no}\:{idea}\:{of} \\ $$$${expressing}\:{all}\:{the}\:{possible}\:{solutions}! \\ $$$${However}\:{I}\:{am}\:{thinking}\:{about}\:{it}... \\ $$

Commented by mr W last updated on 29/Sep/19

thank you for trying sir!

$${thank}\:{you}\:{for}\:{trying}\:{sir}! \\ $$

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.

$${Sir}\:{I}\:{think}\:{infinite}\:{solutions}! \\ $$$${I}'{ll}\:{try}\:{to}\:{write}\:{some}\:{observations} \\ $$$${about}\:{such}\:{triplets}\:{which}\:{satisfy} \\ $$$${your}\:{equation}. \\ $$

Commented by mind is power last updated on 29/Sep/19

suppose x and y coprime  ⇒x+y and xy are coprim   ⇒xy∣z⇒z=xy and x+y=1  now assume x and y are note coprim  let d=gcd(x,y)⇒x=dx′,y=dy′  x′,y′ coprime  ⇒d^2 x′y′=d(x′+y′)z  ⇒dx′y′=(x′+y′)z  x′y′∣z  cause x′+y′ and x′y′ are coprime   and x′y′∣dx′y′=(x′+y′)z  z=x′y′.k  ⇒d=k(x′+y′)  let S={n∈N   n∣d}  for all m∈N^∗  we can Whrite m=p+q withe p,q coprim m=1+(m−1)  k=m withe m∈S  x′+y′=(d/m)∈S  (x′,y′)∈{(a,b)∣a+b=(d/m),and gcd(a,b)=1}  x=dx′=da  y=dx′=db  z=kx′y′=mab

$${suppose}\:{x}\:{and}\:{y}\:{coprime} \\ $$$$\Rightarrow{x}+{y}\:{and}\:{xy}\:{are}\:{coprim}\: \\ $$$$\Rightarrow{xy}\mid{z}\Rightarrow{z}={xy}\:{and}\:{x}+{y}=\mathrm{1} \\ $$$${now}\:{assume}\:{x}\:{and}\:{y}\:{are}\:{note}\:{coprim} \\ $$$${let}\:{d}={gcd}\left({x},{y}\right)\Rightarrow{x}={dx}',{y}={dy}' \\ $$$${x}',{y}'\:{coprime} \\ $$$$\Rightarrow{d}^{\mathrm{2}} {x}'{y}'={d}\left({x}'+{y}'\right){z} \\ $$$$\Rightarrow{dx}'{y}'=\left({x}'+{y}'\right){z} \\ $$$${x}'{y}'\mid{z}\:\:{cause}\:{x}'+{y}'\:{and}\:{x}'{y}'\:{are}\:{coprime}\: \\ $$$${and}\:{x}'{y}'\mid{dx}'{y}'=\left({x}'+{y}'\right){z} \\ $$$${z}={x}'{y}'.{k} \\ $$$$\Rightarrow{d}={k}\left({x}'+{y}'\right) \\ $$$${let}\:{S}=\left\{{n}\in\mathbb{N}\:\:\:{n}\mid{d}\right\} \\ $$$${for}\:{all}\:{m}\in\mathbb{N}^{\ast} \:{we}\:{can}\:{Whrite}\:{m}={p}+{q}\:{withe}\:{p},{q}\:{coprim}\:{m}=\mathrm{1}+\left({m}−\mathrm{1}\right) \\ $$$${k}={m}\:{withe}\:{m}\in{S} \\ $$$${x}'+{y}'=\frac{{d}}{{m}}\in{S} \\ $$$$\left({x}',{y}'\right)\in\left\{\left({a},{b}\right)\mid{a}+{b}=\frac{{d}}{{m}},{and}\:{gcd}\left({a},{b}\right)=\mathrm{1}\right\} \\ $$$${x}={dx}'={da} \\ $$$${y}={dx}'={db} \\ $$$${z}={kx}'{y}'={mab} \\ $$$$ \\ $$

Commented by Prithwish sen last updated on 30/Sep/19

  Given  xy=xz+yz  ⇒ 2xy−2xz−2yz = 0 ......(i)  and 2xy+2xz+2yz=(x+y+z)^2 −(x^2 +y^2 +z^2 ).....(ii)  Adding (i) and (ii)  ⇒xy=(((x+y+z)^2 −(x^2 +y^2 +z^2 ))/4) .....(iii)  Now there are three possibilities                        x            y           z  Case I     even      odd     even  Case II   even        even   odd  CaseIII   even     even    even  Now here I am considering only Case I   Let x= 2m  y= 2n+1  z = 2p       ⇒  p = ((m(2n+1))/(2(m+n)+1))  ⇒ 2mp+2np+p = 2mn +m ........(iv)  Now from (iii)    xy = (((2m+2n+2p+1)^2 −{4m^2 +(2n+1)^2 +4p^2 })/4)             = ((8mn+8mp+8np+4m+4p)/4)             =( 2mp+2np+p)+2mn+m = 4mn +2m     Now ∵ p is an integer  ∴ p = n+ ((m−2n^2 −n)/(2m+2n+1))  ⇒m−2n^2 −n=0  ⇒ n = (((√(1+8m))−1)/4)  = (((1+4q)−1)/4) = q  putting  m = 2q^2 +q  p =(((2q^2 +q)(2q+1))/(2{(2q^2 +q)+q}+1)) = ((q(2q+1)^2 )/(2q(2q+1)+2q+1))    =q  ∴ x = 2q(2q+1) , y =2q+1 and z = 2q         q        x           y       z           1       6          3       2           2      20         5        4           3      42         7        6            4      72        9       8         and so on .........  m,n,p ,q are integers  Mr. W Sir and Rasheed Sir please give your  valuable suggestions.Sir I  make the whole thing   only dependent on q. By putting q=1,2,...  and you get the result.

$$ \\ $$$$\mathrm{Given} \\ $$$$\mathrm{xy}=\mathrm{xz}+\mathrm{yz} \\ $$$$\Rightarrow\:\mathrm{2xy}−\mathrm{2xz}−\mathrm{2yz}\:=\:\mathrm{0}\:......\left(\mathrm{i}\right) \\ $$$$\mathrm{and}\:\mathrm{2xy}+\mathrm{2xz}+\mathrm{2yz}=\left(\mathrm{x}+\mathrm{y}+\mathrm{z}\right)^{\mathrm{2}} −\left(\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} \right).....\left(\mathrm{ii}\right) \\ $$$$\mathrm{Adding}\:\left(\mathrm{i}\right)\:\mathrm{and}\:\left(\mathrm{ii}\right) \\ $$$$\Rightarrow\mathrm{xy}=\frac{\left(\mathrm{x}+\mathrm{y}+\mathrm{z}\right)^{\mathrm{2}} −\left(\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} \right)}{\mathrm{4}}\:.....\left(\mathrm{iii}\right) \\ $$$$\boldsymbol{\mathrm{Now}}\:\boldsymbol{\mathrm{there}}\:\boldsymbol{\mathrm{are}}\:\boldsymbol{\mathrm{three}}\:\boldsymbol{\mathrm{possibilities}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{x}}\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{y}}\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{z}} \\ $$$$\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{I}}\:\:\:\:\:\boldsymbol{\mathrm{even}}\:\:\:\:\:\:\boldsymbol{\mathrm{odd}}\:\:\:\:\:\boldsymbol{\mathrm{even}} \\ $$$$\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{I}}\mathrm{I}\:\:\:\boldsymbol{\mathrm{even}}\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{even}}\:\:\:\boldsymbol{\mathrm{odd}} \\ $$$$\boldsymbol{\mathrm{CaseIII}}\:\:\:\boldsymbol{\mathrm{even}}\:\:\:\:\:\boldsymbol{\mathrm{even}}\:\:\:\:\boldsymbol{\mathrm{even}} \\ $$$$\boldsymbol{\mathrm{Now}}\:\boldsymbol{\mathrm{here}}\:\boldsymbol{\mathrm{I}}\:\boldsymbol{\mathrm{am}}\:\boldsymbol{\mathrm{considering}}\:\boldsymbol{\mathrm{only}}\:\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{I}}\: \\ $$$$\mathrm{Let}\:\mathrm{x}=\:\mathrm{2m}\:\:\mathrm{y}=\:\mathrm{2n}+\mathrm{1}\:\:\mathrm{z}\:=\:\mathrm{2p} \\ $$$$\:\:\:\:\:\Rightarrow\:\:\mathrm{p}\:=\:\frac{\mathrm{m}\left(\mathrm{2n}+\mathrm{1}\right)}{\mathrm{2}\left(\mathrm{m}+\mathrm{n}\right)+\mathrm{1}} \\ $$$$\Rightarrow\:\mathrm{2}\boldsymbol{\mathrm{mp}}+\mathrm{2}\boldsymbol{\mathrm{np}}+\boldsymbol{\mathrm{p}}\:=\:\mathrm{2}\boldsymbol{\mathrm{mn}}\:+\boldsymbol{\mathrm{m}}\:........\left(\boldsymbol{\mathrm{iv}}\right) \\ $$$$\mathrm{Now}\:\mathrm{from}\:\left(\mathrm{iii}\right) \\ $$$$\:\:\boldsymbol{\mathrm{xy}}\:=\:\frac{\left(\mathrm{2}\boldsymbol{\mathrm{m}}+\mathrm{2}\boldsymbol{\mathrm{n}}+\mathrm{2}\boldsymbol{\mathrm{p}}+\mathrm{1}\right)^{\mathrm{2}} −\left\{\mathrm{4}\boldsymbol{\mathrm{m}}^{\mathrm{2}} +\left(\mathrm{2}\boldsymbol{\mathrm{n}}+\mathrm{1}\right)^{\mathrm{2}} +\mathrm{4}\boldsymbol{\mathrm{p}}^{\mathrm{2}} \right\}}{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{8}\boldsymbol{\mathrm{mn}}+\mathrm{8}\boldsymbol{\mathrm{mp}}+\mathrm{8}\boldsymbol{\mathrm{np}}+\mathrm{4}\boldsymbol{\mathrm{m}}+\mathrm{4}\boldsymbol{\mathrm{p}}}{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\left(\:\mathrm{2}\boldsymbol{\mathrm{mp}}+\mathrm{2}\boldsymbol{\mathrm{np}}+\boldsymbol{\mathrm{p}}\right)+\mathrm{2}\boldsymbol{\mathrm{mn}}+\boldsymbol{\mathrm{m}}\:=\:\mathrm{4}\boldsymbol{\mathrm{mn}}\:+\mathrm{2}\boldsymbol{\mathrm{m}} \\ $$$$\:\:\:\mathrm{Now}\:\because\:\boldsymbol{\mathrm{p}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{an}}\:\boldsymbol{\mathrm{integer}} \\ $$$$\therefore\:\mathrm{p}\:=\:\mathrm{n}+\:\frac{\mathrm{m}−\mathrm{2n}^{\mathrm{2}} −\mathrm{n}}{\mathrm{2m}+\mathrm{2n}+\mathrm{1}} \\ $$$$\Rightarrow\mathrm{m}−\mathrm{2n}^{\mathrm{2}} −\mathrm{n}=\mathrm{0} \\ $$$$\Rightarrow\:\boldsymbol{\mathrm{n}}\:=\:\frac{\sqrt{\mathrm{1}+\mathrm{8}\boldsymbol{\mathrm{m}}}−\mathrm{1}}{\mathrm{4}}\:\:=\:\frac{\left(\mathrm{1}+\mathrm{4}\boldsymbol{\mathrm{q}}\right)−\mathrm{1}}{\mathrm{4}}\:=\:\boldsymbol{\mathrm{q}}\:\:\boldsymbol{\mathrm{putting}}\:\:\boldsymbol{\mathrm{m}}\:=\:\mathrm{2}\boldsymbol{\mathrm{q}}^{\mathrm{2}} +\boldsymbol{\mathrm{q}} \\ $$$$\mathrm{p}\:=\frac{\left(\mathrm{2}\boldsymbol{\mathrm{q}}^{\mathrm{2}} +\boldsymbol{\mathrm{q}}\right)\left(\mathrm{2}\boldsymbol{\mathrm{q}}+\mathrm{1}\right)}{\mathrm{2}\left\{\left(\mathrm{2}\boldsymbol{\mathrm{q}}^{\mathrm{2}} +\boldsymbol{\mathrm{q}}\right)+\boldsymbol{\mathrm{q}}\right\}+\mathrm{1}}\:=\:\frac{\boldsymbol{\mathrm{q}}\left(\mathrm{2}\boldsymbol{\mathrm{q}}+\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{2}\boldsymbol{\mathrm{q}}\left(\mathrm{2}\boldsymbol{\mathrm{q}}+\mathrm{1}\right)+\mathrm{2}\boldsymbol{\mathrm{q}}+\mathrm{1}} \\ $$$$\:\:=\boldsymbol{\mathrm{q}} \\ $$$$\therefore\:\boldsymbol{\mathrm{x}}\:=\:\mathrm{2}\boldsymbol{\mathrm{q}}\left(\mathrm{2}\boldsymbol{\mathrm{q}}+\mathrm{1}\right)\:,\:\boldsymbol{\mathrm{y}}\:=\mathrm{2}\boldsymbol{\mathrm{q}}+\mathrm{1}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{z}}\:=\:\mathrm{2}\boldsymbol{\mathrm{q}} \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\mathrm{q}}\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{x}}\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{y}}\:\:\:\:\:\:\:\boldsymbol{\mathrm{z}} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{6}\:\:\:\:\:\:\:\:\:\:\mathrm{3}\:\:\:\:\:\:\:\mathrm{2} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{2}\:\:\:\:\:\:\mathrm{20}\:\:\:\:\:\:\:\:\:\mathrm{5}\:\:\:\:\:\:\:\:\mathrm{4} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{3}\:\:\:\:\:\:\mathrm{42}\:\:\:\:\:\:\:\:\:\mathrm{7}\:\:\:\:\:\:\:\:\mathrm{6} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{4}\:\:\:\:\:\:\mathrm{72}\:\:\:\:\:\:\:\:\mathrm{9}\:\:\:\:\:\:\:\mathrm{8} \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{so}}\:\boldsymbol{\mathrm{on}}\:......... \\ $$$$\boldsymbol{\mathrm{m}},\boldsymbol{\mathrm{n}},\boldsymbol{\mathrm{p}}\:,\boldsymbol{\mathrm{q}}\:\boldsymbol{\mathrm{are}}\:\boldsymbol{\mathrm{integers}} \\ $$$$\boldsymbol{\mathrm{Mr}}.\:\boldsymbol{\mathrm{W}}\:\boldsymbol{\mathrm{Sir}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{Rasheed}}\:\boldsymbol{\mathrm{Sir}}\:\boldsymbol{\mathrm{please}}\:\boldsymbol{\mathrm{give}}\:\boldsymbol{\mathrm{your}} \\ $$$$\boldsymbol{\mathrm{valuable}}\:\boldsymbol{\mathrm{suggestions}}.\boldsymbol{\mathrm{Sir}}\:\boldsymbol{\mathrm{I}}\:\:\boldsymbol{\mathrm{make}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{whole}}\:\boldsymbol{\mathrm{thing}}\: \\ $$$$\boldsymbol{\mathrm{only}}\:\boldsymbol{\mathrm{dependent}}\:\boldsymbol{\mathrm{on}}\:\boldsymbol{\mathrm{q}}.\:\boldsymbol{\mathrm{By}}\:\boldsymbol{\mathrm{putting}}\:\boldsymbol{\mathrm{q}}=\mathrm{1},\mathrm{2},... \\ $$$$\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{you}}\:\boldsymbol{\mathrm{get}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{result}}. \\ $$

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^2 ab  don′t satisfy xy=(x+y)z. besides  how to get a,b,d,k?

$${sir}\:{mind}\:{is}\:{power}: \\ $$$${thanks}\:{for}\:{attempting}\:{sir}! \\ $$$${but} \\ $$$${x}={da} \\ $$$${y}={db} \\ $$$${z}={kd}^{\mathrm{2}} {ab} \\ $$$${don}'{t}\:{satisfy}\:{xy}=\left({x}+{y}\right){z}.\:{besides} \\ $$$${how}\:{to}\:{get}\:{a},{b},{d},{k}? \\ $$

Commented by mind is power last updated on 29/Sep/19

i fix i switch z and k

$${i}\:{fix}\:{i}\:{switch}\:{z}\:{and}\:{k}\: \\ $$

Commented by mr W last updated on 29/Sep/19

prithwish sir:  thanks for trying!  with x=2m, y=(((√(1+8m))+1)/2),z=2p  if you take m=2, y≠integer!

$${prithwish}\:{sir}: \\ $$$${thanks}\:{for}\:{trying}! \\ $$$${with}\:{x}=\mathrm{2}{m},\:{y}=\frac{\sqrt{\mathrm{1}+\mathrm{8}{m}}+\mathrm{1}}{\mathrm{2}},{z}=\mathrm{2}{p} \\ $$$${if}\:{you}\:{take}\:{m}=\mathrm{2},\:{y}\neq{integer}! \\ $$

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+2y+z=7.

$${there}\:{are}\:{three}\:{unknowns}\:{x},{y},{z}. \\ $$$${a}\:{general}\:{solution}\:{must}\:{have}\:{two} \\ $$$${parameters}\:{in}\:{the}\:{expression}.\:{this} \\ $$$${is}\:{like}\:{to}\:{find}\:{the}\:{general}\:{integer} \\ $$$${solution}\:{for}\:{x}+\mathrm{2}{y}+{z}=\mathrm{7}. \\ $$

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=(d/m)  and gcd(a,b)=1  a=(d/m)−b  gcd(a,b)=1⇒gcd((d/m),b)  soit b is coprime withe (d/(m )) withe b≤(d/m)  same withe a coprime withe (d/m)  (a,b)∗describe all integer withe a+b=(d/m) withe a and b coprime withe (d/m)  exampl a+b=6  (a,b)∈{(1,5) (5,1)}  Sorry my english not Good

$${d}\:{given}\:{parameter} \\ $$$${z}={dab} \\ $$$${m}\:{divusor}\:{of}\:{d}\: \\ $$$${a}\:{and}\:{b}\:{are}\:{such}\:{a}+{b}=\frac{{d}}{{m}} \\ $$$${and}\:{gcd}\left({a},{b}\right)=\mathrm{1} \\ $$$${a}=\frac{{d}}{{m}}−{b} \\ $$$${gcd}\left({a},{b}\right)=\mathrm{1}\Rightarrow{gcd}\left(\frac{{d}}{{m}},{b}\right) \\ $$$${soit}\:{b}\:{is}\:{coprime}\:{withe}\:\frac{{d}}{{m}\:}\:{withe}\:{b}\leqslant\frac{{d}}{{m}} \\ $$$${same}\:{withe}\:{a}\:{coprime}\:{withe}\:\frac{{d}}{{m}} \\ $$$$\left({a},{b}\right)\ast{describe}\:{all}\:{integer}\:{withe}\:{a}+{b}=\frac{{d}}{{m}}\:{withe}\:{a}\:{and}\:{b}\:{coprime}\:{withe}\:\frac{{d}}{{m}} \\ $$$${exampl}\:{a}+{b}=\mathrm{6} \\ $$$$\left({a},{b}\right)\in\left\{\left(\mathrm{1},\mathrm{5}\right)\:\left(\mathrm{5},\mathrm{1}\right)\right\} \\ $$$${Sorry}\:{my}\:{english}\:{not}\:{Good} \\ $$$$ \\ $$$$ \\ $$

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∈Z^+   x=p(p+q)  y=q(p+q)  z=pq  We can algebraically prove that  (x,y,z)=( p(p+q) , q(p+q) , pq ) is solution  of  xy=(x+y)z  This proves that the solutions are  infinite.  Claim:  ^• If  p & q be coprime then  (x,y,z) will be primitive triple satisfying  xy=(x+y)z  ^• The above formulas can produce    all the possible primitive solutions  ^• Non-primitive solution are merely    integral multiples of primitive solutions.       Your offered example( (8,24,6)) is      is non-primitive solution and its      corresponding primitive solution     is (4,12,3) and this can be produced     by my formulas (put p=1,q=3).  How do you see this claim.please  give a counter example if you don′t  agree.

$${I}\:{have}\:{made}\:{a}\:{formula}\:{which}\:{produce} \\ $$$${a}\:{triplet}\:{satisfying}\:{the}\:{given}\:{equation} \\ $$$${for}\:{every}\:{pair}\:{of}\:{positive}\:{integers}: \\ $$$$\:{Let}\:{p},{q}\in\mathbb{Z}^{+} \\ $$$${x}={p}\left({p}+{q}\right) \\ $$$${y}={q}\left({p}+{q}\right) \\ $$$${z}={pq} \\ $$$${We}\:{can}\:{algebraically}\:{prove}\:{that} \\ $$$$\left({x},{y},{z}\right)=\left(\:{p}\left({p}+{q}\right)\:,\:{q}\left({p}+{q}\right)\:,\:{pq}\:\right)\:{is}\:{solution} \\ $$$${of}\:\:\boldsymbol{{xy}}=\left(\boldsymbol{{x}}+\boldsymbol{{y}}\right)\boldsymbol{{z}} \\ $$$${This}\:{proves}\:{that}\:{the}\:{solutions}\:{are} \\ $$$${infinite}. \\ $$$$\boldsymbol{{Claim}}: \\ $$$$\:^{\bullet} {If}\:\:{p}\:\&\:{q}\:{be}\:{coprime}\:{then} \\ $$$$\left({x},{y},{z}\right)\:{will}\:{be}\:{primitive}\:{triple}\:{satisfying} \\ $$$${xy}=\left({x}+{y}\right){z} \\ $$$$\:^{\bullet} {The}\:{above}\:{formulas}\:{can}\:{produce} \\ $$$$\:\:{all}\:{the}\:{possible}\:{primitive}\:{solutions} \\ $$$$\:^{\bullet} {Non}-{primitive}\:{solution}\:{are}\:{merely} \\ $$$$\:\:{integral}\:{multiples}\:{of}\:{primitive}\:{solutions}. \\ $$$$ \\ $$$$\:\:\:{Your}\:{offered}\:{example}\left(\:\left(\mathrm{8},\mathrm{24},\mathrm{6}\right)\right)\:{is} \\ $$$$\:\:\:\:{is}\:{non}-{primitive}\:{solution}\:{and}\:{its} \\ $$$$\:\:\:\:{corresponding}\:{primitive}\:{solution} \\ $$$$\:\:\:{is}\:\left(\mathrm{4},\mathrm{12},\mathrm{3}\right)\:{and}\:{this}\:{can}\:{be}\:{produced} \\ $$$$\:\:\:{by}\:{my}\:{formulas}\:\left({put}\:{p}=\mathrm{1},{q}=\mathrm{3}\right). \\ $$$${How}\:{do}\:{you}\:{see}\:{this}\:{claim}.{please} \\ $$$${give}\:{a}\:{counter}\:{example}\:{if}\:{you}\:{don}'{t} \\ $$$${agree}. \\ $$

Commented by Prithwish sen last updated on 30/Sep/19

As a whole  for the expression   xy=(x+y)z    x,y,z ∈Z^+   we have  Case I x=2q(2q+1)  y = (2q+1)  z=2q  Case II x = 2q     y = 2q(2q−1)  z = 2q−1  and for Case III no such expression found.                      x              y            z  q=1           6              3             2    −Case I                       2              2             1−Case II  q=2            20            5             4  − Case I                        4             12            3 − Case II   (value of  q=3             42            7             6− Case I       x and y                           6             30           5 − Case II    can be   q=4              72            9             8 − Case I     altered)                            8            56           7 − Case II  and so on......  It confirms that the expression has infinite  number of solutions .

$$\mathrm{As}\:\mathrm{a}\:\mathrm{whole} \\ $$$$\mathrm{for}\:\mathrm{the}\:\mathrm{expression}\: \\ $$$$\boldsymbol{\mathrm{xy}}=\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right)\boldsymbol{\mathrm{z}}\:\:\:\:\boldsymbol{\mathrm{x}},\boldsymbol{\mathrm{y}},\boldsymbol{\mathrm{z}}\:\in\mathbb{Z}^{+} \\ $$$$\mathrm{we}\:\mathrm{have} \\ $$$$\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{I}}\:\boldsymbol{\mathrm{x}}=\mathrm{2}\boldsymbol{\mathrm{q}}\left(\mathrm{2}\boldsymbol{\mathrm{q}}+\mathrm{1}\right)\:\:\boldsymbol{\mathrm{y}}\:=\:\left(\mathrm{2}\boldsymbol{\mathrm{q}}+\mathrm{1}\right)\:\:\boldsymbol{\mathrm{z}}=\mathrm{2}\boldsymbol{\mathrm{q}} \\ $$$$\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{II}}\:\boldsymbol{\mathrm{x}}\:=\:\mathrm{2}\boldsymbol{\mathrm{q}}\:\:\:\:\:\boldsymbol{\mathrm{y}}\:=\:\mathrm{2}\boldsymbol{\mathrm{q}}\left(\mathrm{2}\boldsymbol{\mathrm{q}}−\mathrm{1}\right)\:\:\boldsymbol{\mathrm{z}}\:=\:\mathrm{2}\boldsymbol{\mathrm{q}}−\mathrm{1} \\ $$$$\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{III}}\:\boldsymbol{\mathrm{no}}\:\boldsymbol{\mathrm{such}}\:\boldsymbol{\mathrm{expression}}\:\boldsymbol{\mathrm{found}}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{x}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{y}}\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{z}} \\ $$$$\boldsymbol{\mathrm{q}}=\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\:\:\:\:−\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{I}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}−\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{II}} \\ $$$$\boldsymbol{\mathrm{q}}=\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{20}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{5}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{4}\:\:−\:\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{I}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{4}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{12}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}\:−\:\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{II}}\:\:\:\left(\boldsymbol{\mathrm{value}}\:\boldsymbol{\mathrm{of}}\right. \\ $$$$\boldsymbol{\mathrm{q}}=\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{42}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{7}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{6}−\:\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{I}}\:\:\:\:\:\:\:\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{y}}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{30}\:\:\:\:\:\:\:\:\:\:\:\mathrm{5}\:−\:\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{II}}\:\:\:\:\boldsymbol{\mathrm{can}}\:\boldsymbol{\mathrm{be}}\: \\ $$$$\left.\boldsymbol{\mathrm{q}}=\mathrm{4}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{72}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{9}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{8}\:−\:\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{I}}\:\:\:\:\:\boldsymbol{\mathrm{altered}}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{8}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{56}\:\:\:\:\:\:\:\:\:\:\:\mathrm{7}\:−\:\boldsymbol{\mathrm{Case}}\:\boldsymbol{\mathrm{II}} \\ $$$$\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{so}}\:\boldsymbol{\mathrm{on}}...... \\ $$$$\boldsymbol{\mathrm{It}}\:\boldsymbol{\mathrm{confirms}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{expression}}\:\boldsymbol{\mathrm{has}}\:\boldsymbol{\mathrm{infinite}} \\ $$$$\boldsymbol{\mathrm{number}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{solutions}}\:. \\ $$

Commented by Rasheed.Sindhi last updated on 03/Oct/19

Sir how can we prove that the above  formulas can produce all possible  solutions?                                        &  What′s the sourse of problem?

$${Sir}\:{how}\:{can}\:{we}\:{prove}\:{that}\:{the}\:{above} \\ $$$${formulas}\:{can}\:{produce}\:\boldsymbol{{all}}\:{possible} \\ $$$${solutions}? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\& \\ $$$${What}'{s}\:{the}\:{sourse}\:{of}\:{problem}? \\ $$

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=6  (x,y)=(10,15) is a solution,but  (8,24) is also a solution.

$${that}'{s}\:{a}\:{nice}\:{way}\:{sir}! \\ $$$${from}\:{given}\:{p}\:{and}\:{q}\:{we}\:{can}\:{construct} \\ $$$${a}\:{solution}.\:{but}\:{there}\:{could}\:{be}\:{more} \\ $$$${than}\:{just}\:{one}\:{solution}\:{for}\:{given}\:{p},\:{q}. \\ $$$${e}.{g}.\:{z}=\mathrm{6} \\ $$$$\left({x},{y}\right)=\left(\mathrm{10},\mathrm{15}\right)\:{is}\:{a}\:{solution},{but} \\ $$$$\left(\mathrm{8},\mathrm{24}\right)\:{is}\:{also}\:{a}\:{solution}. \\ $$

Commented by Rasheed.Sindhi last updated on 30/Sep/19

Yes sir,the formula doesn′t cover all  possible solutions.However it assures  that there are infinite solutions.  For composite z, the formula can  give multiple solutions. For example  z=6⇒pq=6⇒(p,q)=(1,6) or (2,3)  ⇒(x,y^� ,z)=(7,42,6) or (10,15,6).  But afterall the formula doesn′t  cover all possible solutions.

$${Yes}\:{sir},{the}\:{formula}\:{doesn}'{t}\:{cover}\:{all} \\ $$$${possible}\:{solutions}.{However}\:{it}\:{assures} \\ $$$${that}\:{there}\:{are}\:{infinite}\:{solutions}. \\ $$$${For}\:{composite}\:{z},\:{the}\:{formula}\:{can} \\ $$$${give}\:{multiple}\:{solutions}.\:{For}\:{example} \\ $$$${z}=\mathrm{6}\Rightarrow{pq}=\mathrm{6}\Rightarrow\left({p},{q}\right)=\left(\mathrm{1},\mathrm{6}\right)\:{or}\:\left(\mathrm{2},\mathrm{3}\right) \\ $$$$\Rightarrow\left({x},\bar {{y}},{z}\right)=\left(\mathrm{7},\mathrm{42},\mathrm{6}\right)\:{or}\:\left(\mathrm{10},\mathrm{15},\mathrm{6}\right). \\ $$$${But}\:{afterall}\:{the}\:{formula}\:{doesn}'{t} \\ $$$${cover}\:{all}\:{possible}\:{solutions}. \\ $$

Commented by Prithwish sen last updated on 30/Sep/19

Mr W. Sir please see my solution I have   fix my error. I think I get a general formula    with a single parameter q.

$$\mathrm{Mr}\:\mathrm{W}.\:\mathrm{Sir}\:\mathrm{please}\:\mathrm{see}\:\mathrm{my}\:\mathrm{solution}\:\mathrm{I}\:\mathrm{have}\: \\ $$$$\mathrm{fix}\:\mathrm{my}\:\mathrm{error}.\:\mathrm{I}\:\mathrm{think}\:\mathrm{I}\:\mathrm{get}\:\mathrm{a}\:\mathrm{general}\:\mathrm{formula}\:\: \\ $$$$\mathrm{with}\:\mathrm{a}\:\mathrm{single}\:\mathrm{parameter}\:\boldsymbol{\mathrm{q}}. \\ $$$$ \\ $$

Commented by Prithwish sen last updated on 30/Sep/19

Sir please waiting for your response.

$$\mathrm{Sir}\:\mathrm{please}\:\mathrm{waiting}\:\mathrm{for}\:\mathrm{your}\:\mathrm{response}. \\ $$

Commented by mr W last updated on 30/Sep/19

it′s a solution. but i think it doesn′t  cover all possible solutions. z=2q  means z is always even. this is not  true, e.g. x=4,y=12,z=3.

$${it}'{s}\:{a}\:{solution}.\:{but}\:{i}\:{think}\:{it}\:{doesn}'{t} \\ $$$${cover}\:{all}\:{possible}\:{solutions}.\:{z}=\mathrm{2}{q} \\ $$$${means}\:{z}\:{is}\:{always}\:{even}.\:{this}\:{is}\:{not} \\ $$$${true},\:{e}.{g}.\:{x}=\mathrm{4},{y}=\mathrm{12},{z}=\mathrm{3}. \\ $$

Commented by Rasheed.Sindhi last updated on 30/Sep/19

Prithwish sen sir  Your ′product machine′ is wonderful!  It produces solutions successfully.  But there are some solutions which  can′t be produced by your machine.  Whereas mr W sir demands such  machine which can produce all possible  solutions!  For example for z=6 your machine  (farmula) produces single triplet  (solution):(42,7,6) by putting q=3.  Whereas many other triplets(which  aren′t come from your machine)  satisfy the equation xy=(x+y)z.  Such as (10,15,6) etc (Can this be  produced by your formula?)

$${Prithwish}\:{sen}\:{sir} \\ $$$${Your}\:'{product}\:{machine}'\:{is}\:{wonderful}! \\ $$$${It}\:{produces}\:{solutions}\:{successfully}. \\ $$$${But}\:{there}\:{are}\:{some}\:{solutions}\:{which} \\ $$$${can}'{t}\:{be}\:{produced}\:{by}\:{your}\:{machine}. \\ $$$${Whereas}\:{mr}\:{W}\:{sir}\:{demands}\:{such} \\ $$$${machine}\:{which}\:{can}\:{produce}\:{all}\:{possible} \\ $$$${solutions}! \\ $$$${For}\:{example}\:{for}\:{z}=\mathrm{6}\:{your}\:{machine} \\ $$$$\left({farmula}\right)\:{produces}\:{single}\:{triplet} \\ $$$$\left({solution}\right):\left(\mathrm{42},\mathrm{7},\mathrm{6}\right)\:{by}\:{putting}\:{q}=\mathrm{3}. \\ $$$${Whereas}\:{many}\:{other}\:{triplets}\left({which}\right. \\ $$$$\left.{aren}'{t}\:{come}\:{from}\:{your}\:{machine}\right) \\ $$$${satisfy}\:{the}\:{equation}\:\boldsymbol{{xy}}=\left(\boldsymbol{{x}}+\boldsymbol{{y}}\right)\boldsymbol{{z}}. \\ $$$${Such}\:{as}\:\left(\mathrm{10},\mathrm{15},\mathrm{6}\right)\:{etc}\:\left({Can}\:{this}\:{be}\right. \\ $$$$\left.{produced}\:{by}\:{your}\:{formula}?\right) \\ $$

Commented by Prithwish sen last updated on 30/Sep/19

Thank you sir.

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{sir}. \\ $$

Commented by mr W last updated on 30/Sep/19

thank you all sirs!

$${thank}\:{you}\:{all}\:{sirs}! \\ $$

Commented by Prithwish sen last updated on 01/Oct/19

Thank you sir.

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{sir}. \\ $$$$ \\ $$

Commented by Rasheed.Sindhi last updated on 01/Oct/19

An Equivalent Expression of   xy=(x+y)z.  xy=(x+y)z⇒xy−yz−zx=0  ⇒xy+y(−z)+(−z)x=0  Now,   (x+y−z)^2 =x^2 +y^2 +z^2 +2(xy+y(−z)+(−z)x)         =x^2 +y^2 +z^2 +2(0)  x^2 +y^2 +z^2 =(x+y−z)^2   This is equivalent to xy=(x+y)z  i-e solutions of both are same.  Note:The idea raised in my mind after  seeing a comment of Prithwish sen sir.

$${An}\:\mathcal{E}{quivalent}\:\mathcal{E}{xpression}\:{of}\: \\ $$$$\boldsymbol{{xy}}=\left(\boldsymbol{{x}}+\boldsymbol{{y}}\right)\boldsymbol{{z}}. \\ $$$$\boldsymbol{{xy}}=\left(\boldsymbol{{x}}+\boldsymbol{{y}}\right)\boldsymbol{{z}}\Rightarrow\boldsymbol{{xy}}−\boldsymbol{{yz}}−\boldsymbol{{zx}}=\mathrm{0} \\ $$$$\Rightarrow\boldsymbol{{xy}}+\boldsymbol{{y}}\left(−\boldsymbol{{z}}\right)+\left(−\boldsymbol{{z}}\right)\boldsymbol{{x}}=\mathrm{0} \\ $$$${Now}, \\ $$$$\:\left(\boldsymbol{{x}}+\boldsymbol{{y}}−\boldsymbol{{z}}\right)^{\mathrm{2}} =\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{z}}^{\mathrm{2}} +\mathrm{2}\left(\boldsymbol{{xy}}+\boldsymbol{{y}}\left(−\boldsymbol{{z}}\right)+\left(−\boldsymbol{{z}}\right)\boldsymbol{{x}}\right) \\ $$$$\:\:\:\:\:\:\:=\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{z}}^{\mathrm{2}} +\mathrm{2}\left(\mathrm{0}\right) \\ $$$$\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{z}}^{\mathrm{2}} =\left(\boldsymbol{{x}}+\boldsymbol{{y}}−\boldsymbol{{z}}\right)^{\mathrm{2}} \\ $$$$\boldsymbol{{This}}\:\boldsymbol{{is}}\:\boldsymbol{{equivalent}}\:\boldsymbol{{to}}\:\boldsymbol{{xy}}=\left(\boldsymbol{{x}}+\boldsymbol{{y}}\right)\boldsymbol{{z}} \\ $$$$\boldsymbol{{i}}-\boldsymbol{{e}}\:\boldsymbol{{solutions}}\:\boldsymbol{{of}}\:\boldsymbol{{both}}\:\boldsymbol{{are}}\:\boldsymbol{{same}}. \\ $$$${Note}:{The}\:{idea}\:{raised}\:{in}\:{my}\:{mind}\:{after} \\ $$$${seeing}\:{a}\:{comment}\:{of}\:{Prithwish}\:{sen}\:{sir}. \\ $$

Commented by Rasheed.Sindhi last updated on 03/Oct/19

mr W sir  By primitive triple I meant a triple  with gcd 1  (9,18,6)=3(3,6,2)  (3,6,2) is primitive triple.  z=2⇒pq=2⇒{p,q}={1,2}  x=1(1+2)=3  y=2(1+2)=6  z=2×1=2

$$\boldsymbol{{mr}}\:\boldsymbol{{W}}\:\boldsymbol{{sir}} \\ $$$${By}\:{primitive}\:{triple}\:{I}\:{meant}\:{a}\:{triple} \\ $$$${with}\:{gcd}\:\mathrm{1} \\ $$$$\left(\mathrm{9},\mathrm{18},\mathrm{6}\right)=\mathrm{3}\left(\mathrm{3},\mathrm{6},\mathrm{2}\right) \\ $$$$\left(\mathrm{3},\mathrm{6},\mathrm{2}\right)\:{is}\:{primitive}\:{triple}. \\ $$$${z}=\mathrm{2}\Rightarrow{pq}=\mathrm{2}\Rightarrow\left\{{p},{q}\right\}=\left\{\mathrm{1},\mathrm{2}\right\} \\ $$$${x}=\mathrm{1}\left(\mathrm{1}+\mathrm{2}\right)=\mathrm{3} \\ $$$${y}=\mathrm{2}\left(\mathrm{1}+\mathrm{2}\right)=\mathrm{6} \\ $$$${z}=\mathrm{2}×\mathrm{1}=\mathrm{2} \\ $$

Commented by mr W last updated on 03/Oct/19

i created this question because i became  unsure to my solution to Q10944.

$${i}\:{created}\:{this}\:{question}\:{because}\:{i}\:{became} \\ $$$${unsure}\:{to}\:{my}\:{solution}\:{to}\:{Q}\mathrm{10944}. \\ $$

Commented by Rasheed.Sindhi last updated on 02/Oct/19

Prithwish sen sir & mind-is-power sir!  Please review my above answer and  give feed back.

$$\boldsymbol{{Prithwish}}\:\boldsymbol{{sen}}\:\boldsymbol{{sir}}\:\&\:\boldsymbol{{mind}}-\boldsymbol{{is}}-\boldsymbol{{power}}\:\boldsymbol{{sir}}! \\ $$$${Please}\:{review}\:{my}\:{above}\:{answer}\:{and} \\ $$$${give}\:{feed}\:{back}. \\ $$

Commented by Rasheed.Sindhi last updated on 02/Oct/19

Sir mr W  Please review my above answer.I′ve  added some material in red lines .  I′m anxiously waiting for your response.

$$\boldsymbol{{Sir}}\:\boldsymbol{{mr}}\:\boldsymbol{{W}} \\ $$$${Please}\:{review}\:{my}\:{above}\:{answer}.{I}'{ve} \\ $$$${added}\:{some}\:{material}\:{in}\:{red}\:{lines}\:. \\ $$$${I}'{m}\:{anxiously}\:{waiting}\:{for}\:{your}\:{response}. \\ $$

Commented by mr W last updated on 02/Oct/19

thanks for all the efforts sir!  you can get (8,24,6) from (4,12,3), but  how can we get (9,18,6) from (4,12,3)?

$${thanks}\:{for}\:{all}\:{the}\:{efforts}\:{sir}! \\ $$$${you}\:{can}\:{get}\:\left(\mathrm{8},\mathrm{24},\mathrm{6}\right)\:{from}\:\left(\mathrm{4},\mathrm{12},\mathrm{3}\right),\:{but} \\ $$$${how}\:{can}\:{we}\:{get}\:\left(\mathrm{9},\mathrm{18},\mathrm{6}\right)\:{from}\:\left(\mathrm{4},\mathrm{12},\mathrm{3}\right)? \\ $$

Commented by mr W last updated on 03/Oct/19

now i see. thanks sir!  the task is now how to express x and y  x=f(p,q)  y=g(p,q)  z=pq  such that x and y cover all possibilities.  can we find the f(p,q) and g(p,q)?

$${now}\:{i}\:{see}.\:{thanks}\:{sir}! \\ $$$${the}\:{task}\:{is}\:{now}\:{how}\:{to}\:{express}\:{x}\:{and}\:{y} \\ $$$${x}={f}\left({p},{q}\right) \\ $$$${y}={g}\left({p},{q}\right) \\ $$$${z}={pq} \\ $$$${such}\:{that}\:{x}\:{and}\:{y}\:{cover}\:{all}\:{possibilities}. \\ $$$${can}\:{we}\:{find}\:{the}\:{f}\left({p},{q}\right)\:{and}\:{g}\left({p},{q}\right)? \\ $$

Commented by Rasheed.Sindhi last updated on 03/Oct/19

x=f(p,q)=p(p+q)  y=g(p,q)=q(p+q)  z=h(p,q)=pq  For a given z=12(say)  z=12⇒pq=12 (for primitive solutions  take p & q coprime)  (p,q)= (1,12),(3,4),(4,3),(12,1)  (1,12): x=1(1+12)=13                  y=12(1+12)=156                 z=12  All possible primitive solutions with  z=12 are: (13,156,12),(21,28,12),  (28,21,12) & (156,13,12)  For primitive solutions with z=12  x & y can′t take other values   (If my claim is right).If x & y have  values other than above, then the  that solution must be non-primitive.  please try counter example:(a,b,12)  such that gcd(a,b,12)=1 and                     ab=12(a+b)

$${x}={f}\left({p},{q}\right)={p}\left({p}+{q}\right) \\ $$$${y}={g}\left({p},{q}\right)={q}\left({p}+{q}\right) \\ $$$${z}={h}\left({p},{q}\right)={pq} \\ $$$${For}\:{a}\:{given}\:{z}=\mathrm{12}\left({say}\right) \\ $$$${z}=\mathrm{12}\Rightarrow{pq}=\mathrm{12}\:\left({for}\:{primitive}\:{solutions}\right. \\ $$$$\left.{take}\:{p}\:\&\:{q}\:{coprime}\right) \\ $$$$\left({p},{q}\right)=\:\left(\mathrm{1},\mathrm{12}\right),\left(\mathrm{3},\mathrm{4}\right),\left(\mathrm{4},\mathrm{3}\right),\left(\mathrm{12},\mathrm{1}\right) \\ $$$$\left(\mathrm{1},\mathrm{12}\right):\:{x}=\mathrm{1}\left(\mathrm{1}+\mathrm{12}\right)=\mathrm{13} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}=\mathrm{12}\left(\mathrm{1}+\mathrm{12}\right)=\mathrm{156} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{z}=\mathrm{12} \\ $$$${All}\:{possible}\:{primitive}\:{solutions}\:{with} \\ $$$${z}=\mathrm{12}\:{are}:\:\left(\mathrm{13},\mathrm{156},\mathrm{12}\right),\left(\mathrm{21},\mathrm{28},\mathrm{12}\right), \\ $$$$\left(\mathrm{28},\mathrm{21},\mathrm{12}\right)\:\&\:\left(\mathrm{156},\mathrm{13},\mathrm{12}\right) \\ $$$${For}\:{primitive}\:{solutions}\:{with}\:{z}=\mathrm{12} \\ $$$${x}\:\&\:{y}\:{can}'{t}\:{take}\:{other}\:{values}\: \\ $$$$\left({If}\:{my}\:{claim}\:{is}\:{right}\right).{If}\:{x}\:\&\:{y}\:{have} \\ $$$${values}\:{other}\:{than}\:{above},\:{then}\:{the} \\ $$$${that}\:{solution}\:{must}\:{be}\:{non}-{primitive}. \\ $$$${please}\:{try}\:{counter}\:{example}:\left({a},{b},\mathrm{12}\right) \\ $$$${such}\:{that}\:{gcd}\left({a},{b},\mathrm{12}\right)=\mathrm{1}\:{and}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{ab}=\mathrm{12}\left({a}+{b}\right) \\ $$

Commented by mr W last updated on 03/Oct/19

thanks again sir!  my question is, for any given p and q,  can we find x=f(p,q) and y=g(p,q) which  produce all solutions, primitive and  non−primitive ones?

$${thanks}\:{again}\:{sir}! \\ $$$${my}\:{question}\:{is},\:{for}\:{any}\:{given}\:\boldsymbol{{p}}\:{and}\:\boldsymbol{{q}}, \\ $$$${can}\:{we}\:{find}\:{x}={f}\left({p},{q}\right)\:{and}\:{y}={g}\left({p},{q}\right)\:{which} \\ $$$${produce}\:\boldsymbol{{all}}\:{solutions},\:{primitive}\:{and} \\ $$$${non}−{primitive}\:{ones}? \\ $$

Commented by Rasheed.Sindhi last updated on 03/Oct/19

By adding one additional parameter k:  x=kp(p+q)  y=kq(p+q)  z=kpq  p & q being coprime and k any positive  integer. I think sir these formulas  can cover all primitive as well as  non-primitive solutions.

$${By}\:{adding}\:{one}\:{additional}\:{parameter}\:{k}: \\ $$$${x}={kp}\left({p}+{q}\right) \\ $$$${y}={kq}\left({p}+{q}\right) \\ $$$${z}={kpq} \\ $$$${p}\:\&\:{q}\:{being}\:{coprime}\:{and}\:{k}\:{any}\:{positive} \\ $$$${integer}.\:{I}\:{think}\:{sir}\:{these}\:{formulas} \\ $$$${can}\:{cover}\:\boldsymbol{{all}}\:{primitive}\:{as}\:{well}\:{as} \\ $$$${non}-{primitive}\:{solutions}. \\ $$

Commented by mr W last updated on 03/Oct/19

big big thanks sir!

$${big}\:{big}\:{thanks}\:{sir}! \\ $$

Answered by Rasheed.Sindhi last updated on 02/Oct/19

In one of my comments here I have  shown that   x^2 +y^2 +z^2 =(x+y−z)^2  is equivalent  to xy=(x+y)z.I mean both have  same solutions.  Can this derived form helps us towarxs  the goal(Expressing all possible solutions).  Observe this new form,it resembles  a^2 +b^2 +c^2 =d^2 .And for this there are  already established formulas:  a∈O;  m,n,p,q∈{0,1,2,...};  (m,n,p,q)=1; m+n+p+q)∈O  For primitive  quadruples   (a,b,c,d):  a=m^2 +n^2 −p^2 −q^2   b=2(mq+np)  c=2(nq−mp),  d=m^2 +n^2 +p^2 +q^2     Here d=x+y−z=m^2 +n^2 +p^2 +q^2   Where {x,y,z}={a,b,c } in any order.  z=a=m^2 +n^2 −p^2 −q^2  (z is odd)      d=(2mq+2np)+(2nq−2mp)−(m^2 +n^2 −p^2 −q^2 )                     =m^2 +n^2 +p^2 +q^2    mq+np+nq−mp−m^2 −n^2 =0   q(m+n)+p(n−m)−m^2 −n^2 =0       q=((p(m−n)+m^2 +n^2 )/(m+n))  Now  x=b=2m(((p(m−n)+m^2 +n^2 )/(m+n)))+2np  y=c=2n(((p(m−n)+m^2 +n^2 )/(m+n)))−2mp  z=a=m^2 +n^2 −p^2 −(((p(m−n)+m^2 +n^2 )/(m+n)))^2                                       (x,y,z) is primitive triple  Similarly  z=b=2mq+2np (z∈E)  ....  ...  z=c=2nq−2mp(z∈E)  .....  These formulas or not of practical  use.Because it′s very difficult to  choose such m,n & p that q is integer  and (m,n,p,q)=1 & other conditions  fulfill.           Imractical farmulas

$${In}\:{one}\:{of}\:{my}\:{comments}\:{here}\:{I}\:{have} \\ $$$${shown}\:{that}\: \\ $$$$\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{z}}^{\mathrm{2}} =\left(\boldsymbol{{x}}+\boldsymbol{{y}}−\boldsymbol{{z}}\right)^{\mathrm{2}} \:\boldsymbol{{is}}\:\boldsymbol{{equivalent}} \\ $$$$\boldsymbol{{to}}\:\boldsymbol{{xy}}=\left(\boldsymbol{{x}}+\boldsymbol{{y}}\right)\boldsymbol{{z}}.\boldsymbol{{I}}\:\boldsymbol{{mean}}\:\boldsymbol{{both}}\:\boldsymbol{{have}} \\ $$$$\boldsymbol{{same}}\:\boldsymbol{{solutions}}. \\ $$$${Can}\:{this}\:{derived}\:{form}\:{helps}\:{us}\:{towarxs} \\ $$$${the}\:{goal}\left(\mathcal{E}{xpressing}\:{all}\:{possible}\:{solutions}\right). \\ $$$${Observe}\:{this}\:{new}\:{form},{it}\:{resembles} \\ $$$$\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{b}}^{\mathrm{2}} +\boldsymbol{{c}}^{\mathrm{2}} =\boldsymbol{{d}}^{\mathrm{2}} .{And}\:{for}\:{this}\:{there}\:{are} \\ $$$${already}\:{established}\:{formulas}: \\ $$$$\boldsymbol{{a}}\in\mathbb{O};\:\:\boldsymbol{{m}},\boldsymbol{{n}},\boldsymbol{{p}},\boldsymbol{{q}}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},...\right\}; \\ $$$$\left.\left(\boldsymbol{{m}},\boldsymbol{{n}},\boldsymbol{{p}},\boldsymbol{{q}}\right)=\mathrm{1};\:\boldsymbol{{m}}+\boldsymbol{{n}}+\boldsymbol{{p}}+\boldsymbol{{q}}\right)\in\mathbb{O} \\ $$$$\boldsymbol{{For}}\:\boldsymbol{{primitive}}\:\:\boldsymbol{{quadruples}}\: \\ $$$$\left(\boldsymbol{{a}},\boldsymbol{{b}},\boldsymbol{{c}},\boldsymbol{{d}}\right): \\ $$$$\boldsymbol{{a}}=\boldsymbol{{m}}^{\mathrm{2}} +\boldsymbol{{n}}^{\mathrm{2}} −\boldsymbol{{p}}^{\mathrm{2}} −\boldsymbol{{q}}^{\mathrm{2}} \\ $$$$\boldsymbol{{b}}=\mathrm{2}\left(\boldsymbol{{mq}}+\boldsymbol{{np}}\right) \\ $$$$\boldsymbol{{c}}=\mathrm{2}\left(\boldsymbol{{nq}}−\boldsymbol{{mp}}\right), \\ $$$$\boldsymbol{{d}}=\boldsymbol{{m}}^{\mathrm{2}} +\boldsymbol{{n}}^{\mathrm{2}} +\boldsymbol{{p}}^{\mathrm{2}} +\boldsymbol{{q}}^{\mathrm{2}} \\ $$$$ \\ $$$${Here}\:\boldsymbol{{d}}=\boldsymbol{{x}}+\boldsymbol{{y}}−\boldsymbol{{z}}=\boldsymbol{{m}}^{\mathrm{2}} +\boldsymbol{{n}}^{\mathrm{2}} +\boldsymbol{{p}}^{\mathrm{2}} +\boldsymbol{{q}}^{\mathrm{2}} \\ $$$${Where}\:\left\{\boldsymbol{{x}},\boldsymbol{{y}},\boldsymbol{{z}}\right\}=\left\{\boldsymbol{{a}},\boldsymbol{{b}},\boldsymbol{{c}}\:\right\}\:{in}\:{any}\:{order}. \\ $$$$\boldsymbol{{z}}=\boldsymbol{{a}}=\boldsymbol{{m}}^{\mathrm{2}} +\boldsymbol{{n}}^{\mathrm{2}} −\boldsymbol{{p}}^{\mathrm{2}} −\boldsymbol{{q}}^{\mathrm{2}} \:\left(\boldsymbol{{z}}\:\boldsymbol{{is}}\:\boldsymbol{{odd}}\right) \\ $$$$\:\:\:\:\boldsymbol{{d}}=\left(\mathrm{2}\boldsymbol{{mq}}+\mathrm{2}\boldsymbol{{np}}\right)+\left(\mathrm{2}\boldsymbol{{nq}}−\mathrm{2}\boldsymbol{{mp}}\right)−\left(\boldsymbol{{m}}^{\mathrm{2}} +\boldsymbol{{n}}^{\mathrm{2}} −\boldsymbol{{p}}^{\mathrm{2}} −\boldsymbol{{q}}^{\mathrm{2}} \right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\boldsymbol{{m}}^{\mathrm{2}} +\boldsymbol{{n}}^{\mathrm{2}} +\boldsymbol{{p}}^{\mathrm{2}} +\boldsymbol{{q}}^{\mathrm{2}} \\ $$$$\:\boldsymbol{{mq}}+\boldsymbol{{np}}+\boldsymbol{{nq}}−\boldsymbol{{mp}}−\boldsymbol{{m}}^{\mathrm{2}} −\boldsymbol{{n}}^{\mathrm{2}} =\mathrm{0} \\ $$$$\:\boldsymbol{{q}}\left(\boldsymbol{{m}}+\boldsymbol{{n}}\right)+\boldsymbol{{p}}\left(\boldsymbol{{n}}−\boldsymbol{{m}}\right)−\boldsymbol{{m}}^{\mathrm{2}} −\boldsymbol{{n}}^{\mathrm{2}} =\mathrm{0} \\ $$$$\:\:\:\:\:\boldsymbol{{q}}=\frac{\boldsymbol{{p}}\left(\boldsymbol{{m}}−\boldsymbol{{n}}\right)+\boldsymbol{{m}}^{\mathrm{2}} +\boldsymbol{{n}}^{\mathrm{2}} }{\boldsymbol{{m}}+\boldsymbol{{n}}} \\ $$$${Now} \\ $$$$\boldsymbol{{x}}=\boldsymbol{{b}}=\mathrm{2}\boldsymbol{{m}}\left(\frac{\boldsymbol{{p}}\left(\boldsymbol{{m}}−\boldsymbol{{n}}\right)+\boldsymbol{{m}}^{\mathrm{2}} +\boldsymbol{{n}}^{\mathrm{2}} }{\boldsymbol{{m}}+\boldsymbol{{n}}}\right)+\mathrm{2}\boldsymbol{{np}} \\ $$$$\boldsymbol{{y}}=\boldsymbol{{c}}=\mathrm{2}\boldsymbol{{n}}\left(\frac{\boldsymbol{{p}}\left(\boldsymbol{{m}}−\boldsymbol{{n}}\right)+\boldsymbol{{m}}^{\mathrm{2}} +\boldsymbol{{n}}^{\mathrm{2}} }{\boldsymbol{{m}}+\boldsymbol{{n}}}\right)−\mathrm{2}\boldsymbol{{mp}} \\ $$$$\boldsymbol{{z}}=\boldsymbol{{a}}=\boldsymbol{{m}}^{\mathrm{2}} +\boldsymbol{{n}}^{\mathrm{2}} −\boldsymbol{{p}}^{\mathrm{2}} −\left(\frac{\boldsymbol{{p}}\left(\boldsymbol{{m}}−\boldsymbol{{n}}\right)+\boldsymbol{{m}}^{\mathrm{2}} +\boldsymbol{{n}}^{\mathrm{2}} }{\boldsymbol{{m}}+\boldsymbol{{n}}}\right)^{\mathrm{2}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\left(\boldsymbol{{x}},\boldsymbol{{y}},\boldsymbol{{z}}\right)\:\boldsymbol{{is}}\:\boldsymbol{{primitive}}\:\boldsymbol{{triple}} \\ $$$${Similarly} \\ $$$$\boldsymbol{{z}}=\boldsymbol{{b}}=\mathrm{2}\boldsymbol{{mq}}+\mathrm{2}\boldsymbol{{np}}\:\left(\boldsymbol{{z}}\in\mathbb{E}\right) \\ $$$$.... \\ $$$$... \\ $$$$\boldsymbol{{z}}=\boldsymbol{{c}}=\mathrm{2}\boldsymbol{{nq}}−\mathrm{2}\boldsymbol{{mp}}\left(\boldsymbol{{z}}\in\mathbb{E}\right) \\ $$$$..... \\ $$$${These}\:{formulas}\:{or}\:{not}\:{of}\:{practical} \\ $$$${use}.{Because}\:{it}'{s}\:{very}\:{difficult}\:{to} \\ $$$${choose}\:{such}\:{m},{n}\:\&\:{p}\:{that}\:{q}\:{is}\:{integer} \\ $$$${and}\:\left({m},{n},{p},{q}\right)=\mathrm{1}\:\&\:{other}\:{conditions} \\ $$$${fulfill}. \\ $$$$\:\:\:\:\:\:\:\:\:\boldsymbol{{Imractical}}\:\boldsymbol{{farmulas}} \\ $$

Commented by Rasheed.Sindhi last updated on 01/Oct/19

I hope these formulas cover all   possible primitive solutions.Non-  primitive solutions are integral   multiple of primitive solutions.

$${I}\:{hope}\:{these}\:{formulas}\:{cover}\:{all}\: \\ $$$${possible}\:{primitive}\:{solutions}.{Non}- \\ $$$${primitive}\:{solutions}\:{are}\:{integral}\: \\ $$$${multiple}\:{of}\:{primitive}\:{solutions}. \\ $$

Commented by Prithwish sen last updated on 01/Oct/19

Sir the primitive quadruples which you   derived from the lebesgue′s identity. Are   you sure that it covers all the possible   primitive quadruples ?

$$\mathrm{Sir}\:\mathrm{the}\:\mathrm{primitive}\:\mathrm{quadruples}\:\mathrm{which}\:\mathrm{you}\: \\ $$$$\mathrm{derived}\:\mathrm{from}\:\mathrm{the}\:\mathrm{lebesgue}'\mathrm{s}\:\mathrm{identity}.\:\mathrm{Are}\: \\ $$$$\mathrm{you}\:\mathrm{sure}\:\mathrm{that}\:\mathrm{it}\:\mathrm{covers}\:\mathrm{all}\:\mathrm{the}\:\mathrm{possible}\: \\ $$$$\mathrm{primitive}\:\mathrm{quadruples}\:?\: \\ $$

Commented by Prithwish sen last updated on 01/Oct/19

I think these are the nessecary but not the  sufficient condition.

$$\mathrm{I}\:\mathrm{think}\:\mathrm{these}\:\mathrm{are}\:\mathrm{the}\:\mathrm{nessecary}\:\mathrm{but}\:\mathrm{not}\:\mathrm{the} \\ $$$$\mathrm{sufficient}\:\mathrm{condition}. \\ $$

Commented by Rasheed.Sindhi last updated on 01/Oct/19

Commented by Rasheed.Sindhi last updated on 01/Oct/19

“Thus, all primitivePythagorean quadruplesu  are characterized by Lebsgue′s identity...”  Sir I meant by above that all possible  primitive roots can be derived.May be  I am wrong.

$$``{Thus},\:{all}\:{primitivePythagorean}\:{quadruplesu} \\ $$$${are}\:{characterized}\:{by}\:{Lebsgue}'{s}\:{identity}...'' \\ $$$${Sir}\:{I}\:{meant}\:{by}\:{above}\:{that}\:{all}\:{possible} \\ $$$${primitive}\:{roots}\:{can}\:{be}\:{derived}.{May}\:{be} \\ $$$${I}\:{am}\:{wrong}. \\ $$

Commented by Prithwish sen last updated on 01/Oct/19

Actually sir I have some doubt to clear. By   the way you have done an outstanding work.  Excellent thinking.

$$\mathrm{Actually}\:\mathrm{sir}\:\mathrm{I}\:\mathrm{have}\:\mathrm{some}\:\mathrm{doubt}\:\mathrm{to}\:\mathrm{clear}.\:\mathrm{By}\: \\ $$$$\mathrm{the}\:\mathrm{way}\:\mathrm{you}\:\mathrm{have}\:\mathrm{done}\:\mathrm{an}\:\mathrm{outstanding}\:\mathrm{work}. \\ $$$$\mathrm{Excellent}\:\mathrm{thinking}. \\ $$

Commented by Rasheed.Sindhi last updated on 03/Oct/19

THANKS in advance sir! My email   is below:     rasheedsindhi@yahoo.com

$$\mathcal{THANKS}\:{in}\:{advance}\:{sir}!\:{My}\:{email} \\ $$$$\:{is}\:{below}: \\ $$$$\:\:\:\boldsymbol{{rasheedsindhi}}@\boldsymbol{{yahoo}}.\boldsymbol{{com}} \\ $$

Commented by Prithwish sen last updated on 03/Oct/19

Sir I have already sentyou the pdf. If you   received that please give a feedback.

$$\mathrm{Sir}\:\mathrm{I}\:\mathrm{have}\:\mathrm{already}\:\mathrm{sentyou}\:\mathrm{the}\:\mathrm{pdf}.\:\mathrm{If}\:\mathrm{you}\: \\ $$$$\mathrm{received}\:\mathrm{that}\:\mathrm{please}\:\mathrm{give}\:\mathrm{a}\:\mathrm{feedback}. \\ $$

Commented by Prithwish sen last updated on 03/Oct/19

Sir actually I find something on net regarding  pythagorean quadruples very interesting. I will post it later.   Now I am little busy. So please wait.

$$\mathrm{Sir}\:\mathrm{actually}\:\mathrm{I}\:\mathrm{find}\:\mathrm{something}\:\mathrm{on}\:\mathrm{net}\:\mathrm{regarding} \\ $$$$\mathrm{pythagorean}\:\mathrm{quadruples}\:\mathrm{very}\:\mathrm{interesting}.\:\mathrm{I}\:\mathrm{will}\:\mathrm{post}\:\mathrm{it}\:\mathrm{later}.\: \\ $$$$\mathrm{Now}\:\mathrm{I}\:\mathrm{am}\:\mathrm{little}\:\mathrm{busy}.\:\mathrm{So}\:\mathrm{please}\:\mathrm{wait}. \\ $$

Commented by Prithwish sen last updated on 03/Oct/19

Commented by Prithwish sen last updated on 03/Oct/19

Commented by Prithwish sen last updated on 03/Oct/19

Commented by Rasheed.Sindhi last updated on 03/Oct/19

ThαnX Prithwish sen sir for this  precious information! pl share web  url also!

$$\mathcal{T}{h}\alpha{n}\mathcal{X}\:{Prithwish}\:{sen}\:{sir}\:{for}\:{this} \\ $$$${precious}\:{information}!\:{pl}\:{share}\:{web} \\ $$$${url}\:{also}! \\ $$

Commented by Prithwish sen last updated on 03/Oct/19

Sir I am not so good at using gadgets. If you  send your mail then I can send you the entire  pdf.

$$\mathrm{Sir}\:\mathrm{I}\:\mathrm{am}\:\mathrm{not}\:\mathrm{so}\:\mathrm{good}\:\mathrm{at}\:\mathrm{using}\:\mathrm{gadgets}.\:\mathrm{If}\:\mathrm{you} \\ $$$$\mathrm{send}\:\mathrm{your}\:\mathrm{mail}\:\mathrm{then}\:\mathrm{I}\:\mathrm{can}\:\mathrm{send}\:\mathrm{you}\:\mathrm{the}\:\mathrm{entire} \\ $$$$\mathrm{pdf}. \\ $$

Commented by Rasheed.Sindhi last updated on 03/Oct/19

Sir I recieved the pdf.I′ll study it.  Thanks again.We should think how  can we relate pythagoren quadruples  to the question above.My attempt in  this connection nearly failed...   Pl sir study my first answer again.  I think formulas there can be used  to produce all possible solutions of  xy=(x+y)z

$${Sir}\:{I}\:{recieved}\:{the}\:{pdf}.{I}'{ll}\:{study}\:{it}. \\ $$$${Thanks}\:{again}.{We}\:{should}\:{think}\:{how} \\ $$$${can}\:{we}\:{relate}\:{pythagoren}\:{quadruples} \\ $$$${to}\:{the}\:{question}\:{above}.{My}\:{attempt}\:{in} \\ $$$${this}\:{connection}\:{nearly}\:{failed}... \\ $$$$\:{Pl}\:{sir}\:{study}\:{my}\:{first}\:{answer}\:{again}. \\ $$$${I}\:{think}\:{formulas}\:{there}\:{can}\:{be}\:{used} \\ $$$${to}\:{produce}\:{all}\:{possible}\:{solutions}\:{of} \\ $$$$\boldsymbol{{xy}}=\left(\boldsymbol{{x}}+\boldsymbol{{y}}\right)\boldsymbol{{z}} \\ $$

Commented by Prithwish sen last updated on 03/Oct/19

Excellent work sir. If all the possible   primitive solutions can be produced then  the all non primitive solutions also can be  produced. Great work Sir.

$$\mathrm{Excellent}\:\mathrm{work}\:\mathrm{sir}.\:\mathrm{If}\:\mathrm{all}\:\mathrm{the}\:\mathrm{possible}\: \\ $$$$\mathrm{primitive}\:\mathrm{solutions}\:\mathrm{can}\:\mathrm{be}\:\mathrm{produced}\:\mathrm{then} \\ $$$$\mathrm{the}\:\mathrm{all}\:\mathrm{non}\:\mathrm{primitive}\:\mathrm{solutions}\:\mathrm{also}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{produced}.\:\mathrm{Great}\:\mathrm{work}\:\mathrm{Sir}. \\ $$

Commented by Rasheed.Sindhi last updated on 03/Oct/19

Thanks for Encouraging sir!

$${Thanks}\:{for}\:\mathcal{E}{ncouraging}\:{sir}! \\ $$

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