Monthly Archives: July 2016

To make {(x-y),(y-z),(x-z),(x+y-z), (x+z-y), (y+z-x)}

    Find three positive integers     such that        ,        ,        ,        ,        ,        are all squares                                 … Continue reading

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To make {(a+b-c), (a+c-b), (b+c-a)} all squares, (a,b,c) in AP

    Find three numbers     in arithmetic progression such that (1)    (2)    (3)      ,     ,     where the common difference is   (1)      ………..   (2)      ………..   (3) … Continue reading

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To make {(x+y+z+u),(x+y+z-u),(x+y-z+u),(x-y+z+u),(-x+y+z+u)} squares

    Find four positive integers     such that           Hence,                                                 … Continue reading

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To make {(x±y), (y±z), (z±x)} all squares

    Find three positive integers     such that are all squares.   Here are some solutions: (A, B, C) = (2399057, 2288168, 1873432) (A, B, C) = (4387539232, 3762939168, 2433899232) (A, B, C) = (1189604889857, 680815132832, 418662940768) (A, … Continue reading

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Cube expressible as a sum of consecutive cubes in two distinct ways

    The sum of     consecutive cubes beginning with     is :     The cube     is expressible as a sum of consecutive cubes in two distinct ways:     Can you find another example? … Continue reading

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Sets of 24 consecutive squares whose sum is a square

    Sets of 24 consecutive squares beginning with     whose sum is a square, meaning   There are six infinite families of solutions whose smallest members are :   …………………………………………………….. …………………………………………………….. …………………………………………………….. …………………………………………………….. …………………………………………………….. …………………………………………………….. …………………………………………………….. …………………………………………………….. …………………………………………………….. … Continue reading

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To make {(B-A),(C-A),(C-B),(D-A),(D-B),(D-C)} squares

    To find four positive integers     such that , ,     , ,     ,     are all squares.   Here are 4 sets of solutions:             Can you find other types … Continue reading

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When (X,Y,Z) in geometrical progression {(X-Y), (X-Z), (Y-Z)} squares

    To find three rational numbers in geometrical progression, the difference of any two of which is a square number.     Here’s one possible solution: The integers, form a Pythagorean triple.   and the following integers, are in … Continue reading

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To make {3(4m^4 – n^4), 8(9m^4 – n^4), 15(16m^4 – n^4), 24(25m^4 – n^4)} squares

    To make each of the following expression squares:   and     are positive integers with                                             … Continue reading

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To make {(C^2 – A^2), (C^2 – B^2), (B^2 – A^2)} squares

    Find positive integers     such that ,        , ,        ,        ,        ,        are all squares.   Here are some solutions:   #1 :     (A, B, C)   =   (88642,   … Continue reading

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