Finding number of solutions for equations involving difference of perfect squares - Hemant Malhotra


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    Number of ways in which a natural number can be expressed as difference of two perfect square

    let number = N = x^2 - y^2
    N = (x - y) (x + y)

    Case 1 -
    let N = even number * even number ( product of 2 even number let 4 = 2 * 2)
    so (x - y) (x + y) = even * even
    so x - y = even, x + y = even
    2x = even + even
    so 2x = even (because sum of even + even = even))
    so x = even/2 (which will be an integer)
    so in this case we will get a integral value of x and y

     

    Case 2 -
    N = even * odd
    x^2 - y^2 = even * odd
    (x - y) (x + y) = even * odd
    x - y = even
    x + y = odd
    2x = even + odd
    2x = odd (because even + odd = odd)
    so x = odd/2 ( odd number /2 will not be an integer so in this case we will not an integer value of x and y)

    Case 3 -
    N = odd * odd
    (x - y)(x + y) = odd * odd
    x - y = odd
    x + y = odd
    2x = odd + odd
    2x = even (because odd + odd = even as 3 + 3 = 6)
    so x=even/2
    which will give integral value

    so This is just basic thing that when number is expressed as even * even or odd * odd then only we could find integral solutions and number could be expressed as difference of two perfect square number

    Type 1 : number which is a multiple of 4

    find number of ways in which 20 could be expressed as difference of two perfect square numbers

    case1
    x^2 - y^2 = 20
    (x - y)(x + y) = 20 = 1 * 20 = 2 * 10 = 4 * 5
    case1 - (x - y) (x + y) = 2 * 10 (even * even) so we will find integral value here
    x - y = 2
    x + y = 10
    so x = 6
    y = 4

     

    case2
    (x - y) (x + y) = 4 * 5 ( even * odd not useful for us ))
    so total number of positive values of x and y is 1 (6,4)
    total solution = 4 * positive integral solutions = 4 * 1 = 4
    so total number of solutions = 4

    Why we multiplied by 4 here ???
    x = 6 and y = 4
    so x^2 = 36 and y^2 = 16
    so 36 - 16 = 20
    but if x=-6 and y=4 then also x^2=36 and y^2=16
    if x = 6 and y = -4 (same case)
    if x = -6 and y = -4 (same case)
    so there will be 4 possible integral values.

    DIRECT FORMULA FOR THIS ( number which is multiple of 4)
    Find the number of positive integer solutions of equation x2 – y2 = 20
    solution: (No of factors after dividing by 4) /2.
    Here number=20 and N/4 = 20/4 = 5 and number of factors of 5 = (1+1) = 2
    so ans = 2/2 = 1 ( so positive integer values=1 ) and TOTAL = 4 * 1 = 4

    Type 2 : NUMBER WHICH IS PERFECT SQUARE

    x^2 - y^2 = 16
    ( x - y ) ( x + y ) = 16 = 1 * 16 = 2 * 8 = 4 * 4

    case-1
    (x - y) (x + y) = 1 * 16 ( odd * even case rejected)

    case-2
    (x - y)(x + y) = 2 * 8 (even * even case)
    x - y = 2
    x + y = 8
    so x = 5 and y = 3
    (5,3)

    case-3
    (x - y) (x + y) = 4 * 4
    x - y = 4
    x + y = 4
    so x = 4 and y = 0
    (4,0)
    so total positive solutions here = 1 only because (4,0) is not a positive solution

    ad-250

    Now the other values (5,3) (5,-3)(-5,3)(-5,-3) (4,0)(-4,0)
    so total solutions = 6 look here total number of solutions are not 4 * positive numbers but 4 * positive + 2

    Direct approach
    (No of factors on the right hand side of given after dividing by 4) – 1 }/2.
    example:
    X2 - Y2=16
    DIVIDE 16 BY 4 = 4
    NOW NUMBER OF FACTORS OF 4 ( 22 ) = 2 + 1 = 3
    NOW ( 3 - 1) / 2 = 1 TOTAL POSITIVE SOLUTIONS = 1
    AND TOTAL INTEGRAL = 4 * positive + 2 (in case of perfect square)
    so 4 * 1 + 2 = 6 total integral solutions

    Practice question : x^2 - y^2 = 36 find number of total integral solution [ Answer is 6 ]

    Type 3 : N = 4k+2 form

    let number =18 = 4 * 4 + 2 form
    x^2 - y^2 = 18
    (x - y)(x + y) = 1 * 18 = 2 * 9 = 3 * 6
    so all cases are even * odd so there will be no solution for this

    NOTE- ANY NUMBER OF 4k+2 form can't be expressed as difference of two perfect square

    So x^2 - y^2 = 26 can't be expressed in this form for any integral values of x and y

    Type 4 : When number is Odd

    x^2 - y^2 = 85
    so (x - y)(x + y) = 1 * 85 = 5 * 17
    so odd * odd

    case1
    (x - y)(x + y) = 1 * 85
    x - y = 1
    x + y = 85
    so x = 43
    y = 42

    case2
    (x - y)(x + y) = 5 * 17
    x - y = 5
    x + y = 17
    x=11 ,y=6
    so two positive solutions
    and 2 * 4 = 8 total integral solutions

    DIRECT FORMULA
    (No of factors) /2
    x^2 - y^2 = 85
    85 = 5 * 17 so factors = 2*2 = 4
    so 4/2=2 (number of positive integral values =2)
    and total = 2 * 4 =8

    Some More...

    1) x^2-y^2=124 ( find total number of integral solutions)
    2) find number of ways in which 256 could be expressed as difference of two perfect square numbers .
    3) x^2-y^2=21 ( find number of positive integral solutions)
    4) x^2-y^2=122 ( find number of integral solutions)
    5) x^2-y^2=100 ( find the no of non negative integral solution? )

     

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