Can we define parity & primality for negative numbers?

1st Part: Can we define a negative number as even or odd?

I think it would be yes because any negative number when divided by 2 either leaves remainder of 0 and 1. I need suggestions about it.

2nd Part: The second part is tricky: Can we define a negative number to be prime or Composite?

If I define positive prime number as follows: An positive Integer having exactly two positive distinct factors. Then 2, 3, 5, 7,.... would be prime and 0, 1, 4, 6,.... would not be prime. If we take -2, it would have infinitely distinct factorizations: -2 = (-1)(2); -2 = (1)(-2); -2 = (-1)(-1)(-1)(2);...... and it doesn't obey fundamental theorem of Arithmetic since it isn't unique. Similarly taking prime factorizations of any negative Integer always give multiple factorizations. So, negative numbers couldn't be defined to be prime or composite in the same way as positive Integer greater than 1.

My real question is 'If I want to define "Negative Prime" in a different way, then how can I do it'?

#NumberTheory

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Comments

Yes, you can do all that. If you study ring theory, these things are well understood.

Agnishom Chattopadhyay - 3 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$