When (x^2 + y^2) and (x^2 + y^2 + z^2) are squares

 
 
Solve in integers the system of equations:

    x^2 \; + \; y^2 \; = \; a^2

x^2 \; + \; y^2 \; + \; z^2 \; = \; b^2

 
 

Take any two integers such that   (x, y) = 1   and   x^2 + y^2 = a^2

Now,   ((a^2 - 1)/2)^2 + a^2 \; = \; ((a^2 + 1)/2)^2

Thus,   z^2 \; = \; ((a^2 - 1)/2)^2

and,   b^2 \; = \; ((a^2 + 1)/2)^2

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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2 Responses to When (x^2 + y^2) and (x^2 + y^2 + z^2) are squares

  1. paul says:

    There will be an infinite set of solutions to those. The first equation is just a Pythagorean Triple. The second equation is the sides of a box that has an integer space diagonal where two sides also has an integral diagonal (first equation).

    Here are the first few that has a x <= 100.

    12^2 + 9^2 = 15^2
    12^2 + 9^2 + 8^2 = 17^2

    24^2 + 18^2 = 30^2
    24^2 + 18^2 + 16^2 = 34^2

    28^2 + 21^2 = 35^2
    28^2 + 21^2 + 12^2 = 37^2

    32^2 + 24^2 = 40^2
    32^2 + 24^2 + 9^2 = 41^2

    36^2 + 27^2 = 45^2
    36^2 + 27^2 + 24^2 = 51^2

    48^2 + 36^2 = 60^2
    48^2 + 36^2 + 32^2 = 68^2

    48^2 + 36^2 = 60^2
    48^2 + 36^2 + 25^2 = 65^2

    48^2 + 36^2 = 60^2
    48^2 + 36^2 + 11^2 = 61^2

    56^2 + 42^2 = 70^2
    56^2 + 42^2 + 24^2 = 74^2

    60^2 + 45^2 = 75^2
    60^2 + 45^2 + 40^2 = 85^2

    64^2 + 48^2 = 80^2
    64^2 + 48^2 + 39^2 = 89^2

    64^2 + 48^2 = 80^2
    64^2 + 48^2 + 18^2 = 82^2

    72^2 + 54^2 = 90^2
    72^2 + 54^2 + 48^2 = 102^2

    84^2 + 63^2 = 105^2
    84^2 + 63^2 + 56^2 = 119^2

    84^2 + 63^2 = 105^2
    84^2 + 63^2 + 36^2 = 111^2

    96^2 + 72^2 = 120^2
    96^2 + 72^2 + 64^2 = 136^2

    96^2 + 72^2 = 120^2
    96^2 + 72^2 + 50^2 = 130^2

    96^2 + 72^2 = 120^2
    96^2 + 72^2 + 35^2 = 125^2

    96^2 + 72^2 = 120^2
    96^2 + 72^2 + 27^2 = 123^2

    96^2 + 72^2 = 120^2
    96^2 + 72^2 + 22^2 = 122^2

    Here are a few much bigger ones

    12345678987654321^2 + 10693821372022228^2 = 16333205600950745^2
    12345678987654321^2 + 10693821372022228^2 + 9698053055367012^2 = 18995416243600513^2

    12345678987654321^2 + 10693821372022228^2 = 16333205600950745^2
    12345678987654321^2 + 10693821372022228^2 + 8510345496995160^2 = 18417371845112825^2

    12345678987654321^2 + 10693821372022228^2 = 16333205600950745^2
    12345678987654321^2 + 10693821372022228^2 + 7700727071378352^2 = 18057541433727577^2

    12345678987654321^2 + 10693821372022228^2 = 16333205600950745^2
    12345678987654321^2 + 10693821372022228^2 + 1038075110073036^2 = 16366160366349889^2

    There are 610 solutions just for x = 123456789987654321 and y = 122083913542000240

    Paul.

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