Why do these numbers don't have a gcd?

[cinder plume]

Working in Z[sqrt(-5)], we have the numbers
1 + isqrt(5) and 3 have gcd = 1
but 2 + 2isqrt(5) and 6 do not have a gcd?? Why

[acoustic flame]

1. Wait patiently for a helper to come along.
2. Once someone helps you, say thank you and close the thread with:

+close

3. Feel free to nominate the person for helper of the week in #helper-nominations
4. Do not ping the mods, unless someone is breaking the rules.
5. If you're happy with the help you got here, and the server overall, you can contribute financially as well:

[fresh haven]

@cinder plume I think Euclid's lemma is a quick clean approach

[cinder plume]

Which is that?

[fresh haven]

More specifically, if gcd (ax,bx) exists, then Euclid's lemma says that
if a | bx and gcd(a,b) = 1, then a | x

hence, if you show that Euclid's lemma fails for these elements, then their gcd cannot exist

in your case a = 1 + sqrt(-5), b = 3 and x = 2

assuming gcd(ax,bx) exists, we must have the Euclid lemma

1. Note that ( 1+sqrt(-5) ) (1-sqrt(-5)) = 6 = bx, so a | bx
2. gcd (a,b) = 1

so it must follow that 1+sqrt(-5) | 2

but is this true? @cinder plume

$$ (1+\sqrt{-5})(u+v\sqrt{-5}) = 2 $$

[sacred perch]

aL

[fresh haven]

does this have a solution?

[true rapids]

||some (u, v) = (k,-k) works||

[fresh haven]

u + (u+v)root(-5) - 5v = 2, hence this can only work if u-5v = 2 and u+v = 0. There is no integer solution to this @true rapids