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Thread: Proof, that product of 5 numbers cannot be square of a positive interger

  1. #1
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    Proof, that product of 5 numbers cannot be square of a positive interger

    "Prove that the product of five consecutive numbers cannot be the square of a positive integer."

    I have tried using examples, and it shows that this is true, but how do i prove this? Please show working/reasoning if possible.

    Thanks
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  2. #2
    Super Member Bacterius's Avatar
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    Hello,
    easy enough. Given five consecutive numbers $\displaystyle a$, $\displaystyle b$, $\displaystyle c$, $\displaystyle d$ and $\displaystyle e$, for their product to be a square they need to combine all their prime factors with even powers. For instance, taking only two numbers for simplicity, for their product to be a square we need for instance $\displaystyle 7 \times 7 = 7^2$ or $\displaystyle 7^4 \times 3^2 = (49 \times 3)^2$, and $\displaystyle 7 \times 3 = 7^1 \times 3^1$ is not a square because every prime factor doesn't have an even power.

    Now assume some prime number greater than three divides $\displaystyle a$. It obviously doesn't divide $\displaystyle a + 1 = b$, $\displaystyle a + 2 = c$, $\displaystyle a + 3 = d$, $\displaystyle a + 4 = e$. Therefore, this prime factor is only present once in the product $\displaystyle abcde$, hence has an odd exponent (or power) ($\displaystyle 1$) and thus $\displaystyle abcde$ is not a square.

    Now just validate the statement with the two prime numbers smaller or equal to three, namely 2 and 3 (just take $\displaystyle a = 2$ and $\displaystyle a = 3$, since any multiple is suitable), to conclude the proof. Oh, and also validate for $\displaystyle a = 1$ and $\displaystyle a = 0$ ! And for negative numbers, you only have to check up to $\displaystyle a = -4$, because the product of five negative numbers is a negative number which cannot be a square !

    Does it make sense ?
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  3. #3
    Super Member Bacterius's Avatar
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    I just noticed a flaw in my proof. I'll correct it here : Note that if $\displaystyle p^2$, $\displaystyle p$ being a prime number greater than three, divides $\displaystyle a$, then $\displaystyle abcde$ has $\displaystyle p$ with an even power. However, if $\displaystyle a = p^2$ with $\displaystyle p$ prime and $\displaystyle p > 3$, then neither $\displaystyle a + 1, a + 2, a + 3, a + 4$ can be squares, applying the same reasoning and introducing the linear increase of the difference between two consecutive squares.

    Does ... does it make sense ?
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  4. #4
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    Hello, cedricc!

    Prove that the product of five consecutive integers
    cannot be the square of a positive integer.

    Let the five consecutive integers be: .$\displaystyle n-2,\:n-1,\:n,\:n+1,\:n+2$


    Their product is: .$\displaystyle P \:=\:(n-2)(n-1)n(n+1)(n+2) \;=\;n(n^2-1)(n^2-4) \;=\;n(n^4-5n+4) $


    Assume $\displaystyle P$ is a square.

    Since $\displaystyle P$ contains the factor $\displaystyle n$, it must contain the factor $\displaystyle n^2$.
    . . That is, $\displaystyle n^4-5n^2+4$ has a factor of $\displaystyle n$ . . . and we can factor it out.

    Hence: .$\displaystyle P \;=\;n^2\left(n^3-5n + \frac{4}{n}\right)$


    Since $\displaystyle P$ is an integer, the cubic expression must be an integer.

    . . Then $\displaystyle \frac{4}{n}$ must be an integer . . . $\displaystyle n$ is a factor of 4.

    Hence: .$\displaystyle n \:=\:\pm1,\:\pm2,\:\pm4$


    And we have: .$\displaystyle \begin{array}{|c|c|} n & P \\ \hline \pm1 & 0 \\ \pm2 & 0 \\ \pm4 & \pm720 \end{array}$ . None of these is the square of a positive integer.

    . . Q.E.D.

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  5. #5
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    2 Soroban
    Imho this statement: "Since is an integer, the cubic expression must be an integer" is wrong. For instans, P=11, $\displaystyle {n^2} = 9
    $ then
    "cubic expression" must be $\displaystyle \frac{{11}}{9}
    $
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  6. #6
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    Quote Originally Posted by Soroban View Post

    Assume $\displaystyle P$ is a square.

    Since $\displaystyle P$ contains the factor $\displaystyle n$, it must contain the factor $\displaystyle n^2$.

    This isn't true. For example, consider the perfect square 36. It is divisible by 18, but not by 18^2.

    If n is prime, that is a different matter.
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