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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- Feb 16th 2010, 09:14 PMbirdmwWhy 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) - Feb 16th 2010, 09:40 PMBacterius
Hello,

Quote:

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

Quote:

why isn't x^2-5 divisible by three? (x is an integer)

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

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

__Final conclusion__: $\displaystyle 3n + 5$ cannot be a perfect square.

Does that make sense ? :) - Feb 16th 2010, 09:55 PMbirdmw
thanks! I need to find out what mod / modulo are now (Thinking)

- Feb 16th 2010, 09:58 PMBacterius
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 :)