Can the sum, difference and product of 2 numbers be perfect squares?

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If we take 2 numbers $x$ and $y$ such that $x>y>0$ and , can $x + y$, $x - y$ and $xy$ all be perfect squares?










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  • Is this on topic? Feels like a math question
    – Dr Xorile
    1 hour ago










  • No I made it myself but could not solve it.
    – Magic turtle
    1 hour ago
















2














If we take 2 numbers $x$ and $y$ such that $x>y>0$ and , can $x + y$, $x - y$ and $xy$ all be perfect squares?










share|improve this question









New contributor




Magic turtle is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




















  • Is this on topic? Feels like a math question
    – Dr Xorile
    1 hour ago










  • No I made it myself but could not solve it.
    – Magic turtle
    1 hour ago














2












2








2







If we take 2 numbers $x$ and $y$ such that $x>y>0$ and , can $x + y$, $x - y$ and $xy$ all be perfect squares?










share|improve this question









New contributor




Magic turtle is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











If we take 2 numbers $x$ and $y$ such that $x>y>0$ and , can $x + y$, $x - y$ and $xy$ all be perfect squares?







mathematics number-theory algebra






share|improve this question









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Magic turtle is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|improve this question









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Check out our Code of Conduct.









share|improve this question




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edited 2 hours ago







Magic turtle













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asked 3 hours ago









Magic turtleMagic turtle

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Magic turtle is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Magic turtle is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












  • Is this on topic? Feels like a math question
    – Dr Xorile
    1 hour ago










  • No I made it myself but could not solve it.
    – Magic turtle
    1 hour ago


















  • Is this on topic? Feels like a math question
    – Dr Xorile
    1 hour ago










  • No I made it myself but could not solve it.
    – Magic turtle
    1 hour ago
















Is this on topic? Feels like a math question
– Dr Xorile
1 hour ago




Is this on topic? Feels like a math question
– Dr Xorile
1 hour ago












No I made it myself but could not solve it.
– Magic turtle
1 hour ago




No I made it myself but could not solve it.
– Magic turtle
1 hour ago










2 Answers
2






active

oldest

votes


















4














The answer is




that they cannot.




Suppose $x+y=a^2$ and $x-y=b^2$. Then




$x,y=frac{a^2pm b^2}2$ and therefore $4xy=a^4-b^4$. So if this too is a square we have $a^4-b^4=c^2$. Now see https://math.stackexchange.com/questions/153546/solving-x4-y4-z2 where it is shown that there are no solutions to this equation in nonzero integers.







share|improve this answer





























    2














    (Before the OP clarifying that 0 is disallowed)



    Yes.




    x = 4, y = 0; x+y = 4, x-y = 4, xy = 0. (x can be any perfect square, I just decided to use 4)







    share|improve this answer























    • Sorry my fault, I meant to include that y cannot = 0.
      – Magic turtle
      2 hours ago










    • But can x be 0?
      – Ian MacDonald
      1 hour ago










    • If x were 0 y would be less than 0 making x + y negative and a negative number cannot be a perfect square
      – Magic turtle
      1 hour ago











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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    4














    The answer is




    that they cannot.




    Suppose $x+y=a^2$ and $x-y=b^2$. Then




    $x,y=frac{a^2pm b^2}2$ and therefore $4xy=a^4-b^4$. So if this too is a square we have $a^4-b^4=c^2$. Now see https://math.stackexchange.com/questions/153546/solving-x4-y4-z2 where it is shown that there are no solutions to this equation in nonzero integers.







    share|improve this answer


























      4














      The answer is




      that they cannot.




      Suppose $x+y=a^2$ and $x-y=b^2$. Then




      $x,y=frac{a^2pm b^2}2$ and therefore $4xy=a^4-b^4$. So if this too is a square we have $a^4-b^4=c^2$. Now see https://math.stackexchange.com/questions/153546/solving-x4-y4-z2 where it is shown that there are no solutions to this equation in nonzero integers.







      share|improve this answer
























        4












        4








        4






        The answer is




        that they cannot.




        Suppose $x+y=a^2$ and $x-y=b^2$. Then




        $x,y=frac{a^2pm b^2}2$ and therefore $4xy=a^4-b^4$. So if this too is a square we have $a^4-b^4=c^2$. Now see https://math.stackexchange.com/questions/153546/solving-x4-y4-z2 where it is shown that there are no solutions to this equation in nonzero integers.







        share|improve this answer












        The answer is




        that they cannot.




        Suppose $x+y=a^2$ and $x-y=b^2$. Then




        $x,y=frac{a^2pm b^2}2$ and therefore $4xy=a^4-b^4$. So if this too is a square we have $a^4-b^4=c^2$. Now see https://math.stackexchange.com/questions/153546/solving-x4-y4-z2 where it is shown that there are no solutions to this equation in nonzero integers.








        share|improve this answer












        share|improve this answer



        share|improve this answer










        answered 1 hour ago









        Gareth McCaughanGareth McCaughan

        60.6k3151235




        60.6k3151235























            2














            (Before the OP clarifying that 0 is disallowed)



            Yes.




            x = 4, y = 0; x+y = 4, x-y = 4, xy = 0. (x can be any perfect square, I just decided to use 4)







            share|improve this answer























            • Sorry my fault, I meant to include that y cannot = 0.
              – Magic turtle
              2 hours ago










            • But can x be 0?
              – Ian MacDonald
              1 hour ago










            • If x were 0 y would be less than 0 making x + y negative and a negative number cannot be a perfect square
              – Magic turtle
              1 hour ago
















            2














            (Before the OP clarifying that 0 is disallowed)



            Yes.




            x = 4, y = 0; x+y = 4, x-y = 4, xy = 0. (x can be any perfect square, I just decided to use 4)







            share|improve this answer























            • Sorry my fault, I meant to include that y cannot = 0.
              – Magic turtle
              2 hours ago










            • But can x be 0?
              – Ian MacDonald
              1 hour ago










            • If x were 0 y would be less than 0 making x + y negative and a negative number cannot be a perfect square
              – Magic turtle
              1 hour ago














            2












            2








            2






            (Before the OP clarifying that 0 is disallowed)



            Yes.




            x = 4, y = 0; x+y = 4, x-y = 4, xy = 0. (x can be any perfect square, I just decided to use 4)







            share|improve this answer














            (Before the OP clarifying that 0 is disallowed)



            Yes.




            x = 4, y = 0; x+y = 4, x-y = 4, xy = 0. (x can be any perfect square, I just decided to use 4)








            share|improve this answer














            share|improve this answer



            share|improve this answer








            edited 1 hour ago

























            answered 3 hours ago









            Excited RaichuExcited Raichu

            6,11821065




            6,11821065












            • Sorry my fault, I meant to include that y cannot = 0.
              – Magic turtle
              2 hours ago










            • But can x be 0?
              – Ian MacDonald
              1 hour ago










            • If x were 0 y would be less than 0 making x + y negative and a negative number cannot be a perfect square
              – Magic turtle
              1 hour ago


















            • Sorry my fault, I meant to include that y cannot = 0.
              – Magic turtle
              2 hours ago










            • But can x be 0?
              – Ian MacDonald
              1 hour ago










            • If x were 0 y would be less than 0 making x + y negative and a negative number cannot be a perfect square
              – Magic turtle
              1 hour ago
















            Sorry my fault, I meant to include that y cannot = 0.
            – Magic turtle
            2 hours ago




            Sorry my fault, I meant to include that y cannot = 0.
            – Magic turtle
            2 hours ago












            But can x be 0?
            – Ian MacDonald
            1 hour ago




            But can x be 0?
            – Ian MacDonald
            1 hour ago












            If x were 0 y would be less than 0 making x + y negative and a negative number cannot be a perfect square
            – Magic turtle
            1 hour ago




            If x were 0 y would be less than 0 making x + y negative and a negative number cannot be a perfect square
            – Magic turtle
            1 hour ago










            Magic turtle is a new contributor. Be nice, and check out our Code of Conduct.










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