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Math Help - perfect squares

  1. #1
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    perfect squares

    A few problems:

    1) Show that for positive integers n, n^4 + 3n^2 + 1 is never a perfect square.

    2) Let f(n) = 2n^2 + 14n + 25. We see that f(0) = 25 = 5^2. Find two positive integers n such that f(n) is a perfect square.

    3) Find all pairs of positive integers (x,y) such that both x^2 + 3y and y^2 + 3x are perfect squares.
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  2. #2
    Newbie Catherine Morland's Avatar
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    1) n^4+3n^2+1=m^2

    \implies\ n^4+3n^2+(1-m^2)=0

    \implies\ n^2=\frac{-3\pm\sqrt{3^2-4(1-m^2)}}{2}=\frac{-3\pm\sqrt{5+4m^2}}{2}

    5+4m^2 is only a perfect square for m^2=1, which would only give n=0.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by scipa View Post
    A few problems:

    1) Show that for positive integers n, n^4 + 3n^2 + 1 is never a perfect square.

    Suppose that there exists a positive integer n such that: n^4+3n^2+1 is a perfect square k^2.

    Then clearly k>n^2+1, since otherwise k^2 would be less than n^4+3n^2+1.
    Also k<n^2+2, since otherwise k^2 would be greater than n^4+3n^2+1.

    But there are no integers k, such that n^2+1<k<n^2+2.

    RonL
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  4. #4
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    Quote Originally Posted by scipa View Post
    A few problems:

    1) Show that for positive integers n, n^4 + 3n^2 + 1 is never a perfect square.

    2) Let f(n) = 2n^2 + 14n + 25. We see that f(0) = 25 = 5^2. Find two positive integers n such that f(n) is a perfect square.

    3) Find all pairs of positive integers (x,y) such that both x^2 + 3y and y^2 + 3x are perfect squares.
    For 2) it can be show that for each integer m, n=1/4\, \left( 1+\sqrt {2} \right) ^{2\,m-1}+1/4\, \left( 1-\sqrt {2}<br />
 \right) ^{2\,m-1}-7/2 is an integer such that f(n) = 2n^2 + 14n + 25 is a square.
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  5. #5
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    Quote Originally Posted by scipa View Post
    A few problems:

    1) Show that for positive integers n, n^4 + 3n^2 + 1 is never a perfect square.

    2) Let f(n) = 2n^2 + 14n + 25. We see that f(0) = 25 = 5^2. Find two positive integers n such that f(n) is a perfect square.

    3) Find all pairs of positive integers (x,y) such that both x^2 + 3y and y^2 + 3x are perfect squares.
    For problem 3), (1,1), (11,16), (16,11) are solutions. I would be interested to learn how to prove there are no more.
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