/sci/ - Science & Math

>Give me an example of two numbers that share a factor other than one when squared, but before being squared just has 'one' as a factor.
Why do you think there are two such numbers?

>>9703387
literally every prime number because being squared makes the prime number itself a factor

>>9703388
Perhaps there are not.

Lets loosen the restrictions a bit then, find me two numbers (decimals, whole numbers, etc. etc.) that share a factor (whole number) other than one when squared, but before being squared do not share any factors at all.

>>9703389
>literally every prime number because being squared makes the prime number itself a factor
But 2^2 and 3^2 don't share factors.

>>9703399
>find me two numbers (decimals, whole numbers, etc. etc.)
Arbitrary decimals don't have any common, useful notion of "factor", and there exist no such whole numbers.

>what is a least common multiple

>>9703423
>>what is a least common multiple
Irrelevant to the thread.

actually
>what is a least common factor
is what I meant, but unlike you it has been many years since I took algebra II

>>9703387
This is not possible in any UFD (i.e. any context where there is a meaningful notion of divisibility, and where prime factorizations are unique).

>>9703387
>two numbers that share a factor other than one when squared, but before being squared just has 'one' as a factor

This is literally impossible. Only primes have just one as a factor, and when you square primes they only have that same number as a factor (and one).

>>9704964
OP is actually calling for relative primeness here. Your argument still stands.

>>9703387

This is trivially impossible

When you square a number, you dont get any new prime factors. Only the exponents of the prime factors it already has are changed.

In order for them to not be relatively prime after being squared, they would have to have a common factor to begin with.

>>9704964
>Only primes have just one as a factor
Primes also have themselves as a factor.

>>9703387
There are none

Proof: Think

>>9703387

well, which ring?

1+sqrt(-5), 1-sqrt(-5) in ZZ[sqrt(-5)]

isn't it, my love?

>>9703387

I believe that's impossible because when you square a number it has all the same prime squares as before it was squared and therefore the two squared numbers aren't going to gain any additional new factors other than new combinations of the previous prime squares.

>>9707140
yeah, I was wondering if the complex numbers counted as an answer.

>>9703387
Literally 4 and 6.
16 and 36 have 2 and 4 as common factors.

>>9703399
Trivial to prove this is impossible.

6 and 9

>>9703387
1 and 0

>>9703387
ugh use the euler phi function or something

>this thread on gauss' birthday
for shame, guys...

>>9707687
the OP asked for the numbers to only have 1 as a factor before being squared, 4 and 6 have 1 and 2 as a factor, it's not possible.

>>9707744
maybe but nobody's proved it yet
gcd, lcm, phi function, modulus some kinda shit like this should do it fuck me

>>9703387
-5 and 5
/thread

>>9707777