Monthly Archives: May 2016

1/x + 1/y = 1/z, (x, z) = 1

    Let     be positive integer for which ,         Prove that    ,       and       are all squares                                 … Continue reading

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a^4 + 2n(ab)^2 + b^4 = c^4 + 2n(cd)^2 + d^4 …. Part 2

    for   n = 1, 2, 3, …, 10     For   n = 11, 12, 13, …, 100                                                                         … Continue reading

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a^4 + 14(a*b)^2 + b^4 = c^4 + 14(c*d)^2 + d^4

    I like this type of equations, because they can be used to form identities.   Let’s solve for   n = 7 where     are distinct positive integers   For example,     Note that So,   … Continue reading

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Equation : x^2 + y^3 = z^4

        Determine the next values. Establish the recurrence relation.                                

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A^2 + B^2 + C^2 = D^2 + E^2

    Also true is, Let       Let’s take all primitive Pythagorean triples with   ,   for   n = 1                                 … Continue reading

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When x^2 + p*y and y^2 + p*x are squares, p is a small odd prime

    Find positive integers     so that the expressions ,     and are to made squares where   p = 3, 5, 7, 11, 13, 17, 19                       … Continue reading

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(a^2 + b^2 + c^2 + d^2)(p^2 + q^2 + r^2 + s^2) — Part 3

          [ a sum of four squares] Or      [ as a sum of seven squares ]     Can you find positive integers     so that the expressions are to be made squares.       … Continue reading

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(a*p + b*q + c*r), (a*q – b*p), (a*r – c*p), (b*r – c*q) — Part 2

    Can you find positive integers     so that the expressions are to be made squares.     Note that                                     … Continue reading

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When a*b + c*d, a*d – b*c are squares — Part 1

    Can you find positive integers     so that the expressions are to be made squares     Note that I want to produce   For example, …….. ……… ……..                 … Continue reading

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a^2 + ab + b^2 = c^2 + cd + d^2; a^4 + (ab)^2 + b^4 = c^4 + (cd)^2 + d^4

    or equivalently, that is,   For example,         for example,                                      

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