Tag Archives: Diophantine Equation

A^2 = B^3 + C^3

    Solve, given all unknowns are non-zero integers                                                           Advertisements

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x^2 + y^2 + z^2 = 3 xyz

    Find 3 positive integers      and      which satisfy this equation:                            Using this solution, find a different solution by changing one and only one of the numbers. How many solutions does the equation have?   … Continue reading

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Diophantine equations : 2^A – 3^B = 3^D – 2^C

    Find all the solutions to the Diophantine equations                                                        

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a^n + b^n + c^n = x^n + y^n, n = 1,3

    (1)   Find a few solutions to the system    where   a, b, c, x, y   are positive integers.   (2)   Prove that there are infinitely many solutions in positive integers, whose greatest common divisor … Continue reading

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Diophantine equation : y^2 – p = 2^n, where p is a prime number

    The diophantine equation     where     is a prime number   < 100   The only solutions to the diophantine equation     with   y > 0   are given by     (n, y)   =   … Continue reading

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Diophantine equation : 2^a + 2^b + 2^c = x^2

    Consider the Diophantine equation : for     nonnegative integers,                                                      

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Diophantine equation : (a-b)/(a+b) + (b-c)/(b+c) + (c-d)/(c+d) + (d-a)/(d+a) = 0

    Determine integral solutions of the Diophantine equation     where     are distinct   Solutions: ,     ,     If   ,   then     ,     If   ,   then     ,         … Continue reading

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Diophantine equation: (x – y – z)(x – y + z)(x + y – z) = 8 xyz

    Does the Diophantine equation have an infinite number of relatively prime solutions?     Solution:   we have the trivial solutions     and permutations thereof. For other solutions, note that each of the three equations is satisfied … Continue reading

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Diophantine equation: (x – y – z)^3 = 27 xyz

    Are there any integral solutions     of the Diophantine equation other than     or such that   ?     Solution: Let so that   ,   or equivalently   ………..   (1) since   (1) … Continue reading

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Diophantine Eq. : a^2 + b^2 + c^2 = 2d^2 and a^4 + b^4 + c^4 = 2d^4

    Determine all integral solutions of the simultaneous Diophantine equations:                                                  

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