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Math Help - Help to prove Primes

  1. #1
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    Help to prove Primes

    Hello,

    Can anyone please asssist with the following question:

    Prove that a positive integer a > 1 is a square if and only if in the canonical form of a all the exponents of the primes are even integers.

    I would surely appreciate it.
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  2. #2
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    Quote Originally Posted by lahuerita View Post
    Hello,

    Can anyone please asssist with the following question:

    Prove that a positive integer a > 1 is a square if and only if in the canonical form of a all the exponents of the primes are even integers.

    I would surely appreciate it.
    p_1^{a_1}\cdot ... \cdot p_n^{a_n}
    You can write,
    \left(p_1^{a_1/2}\cdot ... p_n^{a_n/2} \right)^2
    Only when a_i are divisible by two.
    Thus, is a_i are all even.
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  3. #3
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    Thanks for the responds that clarifies a lot
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  4. #4
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    Proving the reverse

    How do you think specifically we could prove the only if part?
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  5. #5
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    Quote Originally Posted by lahuerita View Post
    How do you think specifically we could prove the only if part?
    Simple, because "square" means there exists an integer c such that,
    c^2=a.
    Now, no matter what canonical prime decomposition c has c^2 MUST have even prime exponents because of the square (it doubles everything). And then you have that a must also have even exponents because of uniqueness of prime power decomposition.
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