Register to reply

Co-prime of vectors

by phynewb
Tags: coprime, vectors
Share this thread:
Feb13-12, 09:53 AM
P: 13
Hi guys

I have a question about the coprime of two vectors
For two vectors (x1,x2) and (y1,y2).
Given a,b with gcf (a,b)=1 .i.e. relatively prime.
I do the linear combination of two vectors
a(x1,x2)+b(y1,y2)=n(z1,z2) with some common factor n and gcf(z1,z2)=1.
If n=1 for any a,b, two vectors are said co-prime.
I wonder if any criteria to prove two vectors are coprime.
For example, (2,3),(1,3) are not coprime b/c (2,3)+(1,3)=3(1,2).
But (7,3),(2,1) are coprime b/c a(7,3)+b(2,1)=(7a+2b,3a+b) and gcf(7a+2b,3a+b)=gcf(a,3a+b)=gcf(a,b)=1.
Also how to generalize it to vectors with n components?

Thank you
Phys.Org News Partner Science news on
Experts defend operational earthquake forecasting, counter critiques
EU urged to convert TV frequencies to mobile broadband
Sierra Nevada freshwater runoff could drop 26 percent by 2100
Feb13-12, 11:09 AM
P: 144
Hi, I havnt checked the details, but such problems screem for the use of the determinant formed by the x's and y's, and can then also be generalized immediately. I guess the condition is that this determinant has no prime factors, that is being 1 or -1. Any prime factor p would allow a nontrivial relation ax+by=0 over F_p, which then lifts to show a relation with gcd(z_1,z_2)=p. Tell me if that works out.
Feb13-12, 12:45 PM
P: 13
Thanks Norwegian!
I think you are almost right. But (4,-3) and (3,-4) are coprime but with det=7 will be a counter example. If we consider higher dimensions, are (2,-1,2,-1) and (-4,1,4,-1) co-prime? Do you know how to show it rigorously?

Feb13-12, 01:09 PM
P: 144
Co-prime of vectors

(4,-3) + (3,-4) is divisible by 7, so they are not coprime. The sum of your other vectors is divisible by 2, so they are also not coprime. My guesses for generalizations: n vectors in n-space, determinant = 1 or -1. Two vectors in n-space, set of all 2x2 minors coprime, m<n vectors in n-space, all mxm minors coprime.
Feb13-12, 04:14 PM
P: 13
I type the wrong vectors. I consider the two (-1,0,3,-1) and (-3,1,1,0). Are they coprime? I think you are right but I do not know how to prove it. Thank you!
Feb13-12, 09:20 PM
P: 144
Yes, those vectors are coprime. You only need to look at the last two components.

Register to reply

Related Discussions
Co-prime of vectors General Math 1
A prime number which equals prime numbers General Math 10
A formula of prime numbers for interval (q; (q+1)^2), where q is prime number. Linear & Abstract Algebra 0
Prime Numbers in the Diophantine equation q=(n^2+1)/p and p is Prime Linear & Abstract Algebra 5
Efficiency: prime test vs prime generator Linear & Abstract Algebra 14