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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- February 16th 2010, 10:14 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) - February 16th 2010, 10:40 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 . Therefore, , that is, , or, pushing even further, . Note that squares can only be equal to or modulo 3, but we already considered the case when it is equal to 0 modulo 3 (it is divisible by three). So assume , so .

__Conclusion__: is either equal to 1 or 2 modulo 3, and thus cannot be divisible by three.

__Final conclusion__: cannot be a perfect square.

Does that make sense ? :) - February 16th 2010, 10:55 PMbirdmw
thanks! I need to find out what mod / modulo are now (Thinking)

- February 16th 2010, 10:58 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 :)