# Prove by induction

• Feb 17th 2010, 12:58 AM
watzer
Prove by induction
.
• Feb 17th 2010, 01:13 PM
tonio
Quote:

Originally Posted by watzer
Hi! I'm stuck!

How to prove that a equation x + y = n always have a positive solution(x,y >= 0) when n > z?

What is z??

"a Linear Diophantine equation"
(Ex: 69x + 39y = n, when n > 20)

This is false: any solution to this equation will have x or y negative if , for example, n = 21

Tonio

My steps have been:
Solve the equation as u solve diofantic equations:
so u get:
Ex:
x + y = 1 (in my case GCD(x,y) were one)

then i did multiply all solutions with the value EX: 21,22,23,24,25 ( if n > 20 )

and i got diffrent x,y values on all five wich had all one value of K were X,Y where positive.
(ex:x = -110-49k, y = 155 + 69k)

and all solutions over n > z had atleast one solution wich both x and y were positive.

And then i got a pattern similar to this

Ex:
n 21---22----23---24----25
X 50---52----54---56----58
Y 100--105--110--115--120
K 2-----2-----3-----3------4

But have no idea how to prove it with induction.

Is it posible to prove X then prove Y and by that K is proven to consist and always have a solution ?

.
• Feb 17th 2010, 01:18 PM
watzer
Yes true, diden't want to post my equation. But what if a equation have atleast one solution positive when n > 5(random number)

How to prove it ?

Prove the basestep: n = 6

Prove n = p + 1

Can u help me please, sorry for the lack of information on this.
• Feb 17th 2010, 06:57 PM
tonio
Quote:

Originally Posted by watzer
Yes true, diden't want to post my equation. But what if a equation have atleast one solution positive when n > 5(random number)

How to prove it ?

Prove the basestep: n = 6

Prove n = p + 1

Can u help me please, sorry for the lack of information on this.

I can't help since I don't understand what you want...you must try harder to write your questions in a clear way.

For example, if it MUST be that $x,y>0$ , then the two-inknown equation $x+y=n$ has at least a positive solution (meaning that both x,y are positive) iff $n\geq 2$...but I'm not sure this is what you want.

Tonio
• Feb 18th 2010, 02:44 AM
watzer
I want to prove for a diofantic equation (x + y = n, n > value) that there is always a value of K where x and y are non-negative. (>= 0)

is it posible to prove that ?
x: 5p+12k
Y: -2p-5k

will always have atleast a nonnegative solution when p > 43

Do you have msn messenger ? So i can show you my calculations, PM me your messenger, i cant PM yet.
• Feb 18th 2010, 07:18 AM
watzer
Thanks for the help, think i have solved it.