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Math Help - Why can't I be a perfect square?

  1. #1
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    Why can't I be a perfect square?

    why can't 5+3n ever be a perfect square? (n is an integer)

    alternatively.

    why isn't x^2-5 divisible by three? (x is an integer)
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  2. #2
    Super Member Bacterius's Avatar
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    Hello,
    why can't 5+3n ever be a perfect square? (n is an integer)
    First, put the expression in a more familiar form : why can't 3n + 5 ever be a perfect square ? For 3n + 5 to be a perfect square, we need to have : 3n + 5 = a^2 for some integer a, that is, 3n = a^2 - 5. Thus, saying that no integer a satisfies this condition is equivalent to showing that a^2 - 5 is never divisible by three. The "alternative" question is in fact necessary to solve the first question, so see the following :

    why isn't x^2-5 divisible by three? (x is an integer)
    Suppose x is divisible by three, so x^2 is divisible by three, so obviously x^2 - 5 is not divisible by three ( x^2 - 5 \equiv 1 \pmod{3}).
    Suppose x is not divisible by three, so x^2 \neq 0 \pmod{3}. Therefore, x^2 - 5 \neq -5 \pmod{3}, that is, x^2 - 5 \neq 1 \pmod{3}, or, pushing even further, (x^2 \ mod 3) - 1 \neq 1 \pmod{3}. Note that squares can only be equal to 0 or 1 modulo 3, but we already considered the case when it is equal to 0 modulo 3 (it is divisible by three). So assume x^2 \equiv 1 \pmod{3}, so x^2 - 5 \equiv -4 \equiv 2 \pmod{3}.

    Conclusion : x^2 - 5 is either equal to 1 or 2 modulo 3, and thus cannot be divisible by three.

    Final conclusion : 3n + 5 cannot be a perfect square.

    Does that make sense ?
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  3. #3
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    thanks! I need to find out what mod / modulo are now
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  4. #4
    Super Member Bacterius's Avatar
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    Quote Originally Posted by birdmw View Post
    thanks! I need to find out what mod / modulo are now
    It's an awesome tool to study divisibility and modular arithmetic in general. Some links :
    Modular arithmetic - Wikipedia, the free encyclopedia
    Math Forum - Ask Dr. Math

    They really make problem solving quicker and easier, and give a steady working ground in number theory word problems
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