Can all mathmatical facts be proven?

[procyon]

Can all mathematical facts be proven?

Hi,

2 equations give 2 lines that cross a hyperbola and intersect on the x-axis

\(\displaystyle u_1+v_1=u_2-v_2\)

\(\displaystyle u_1v_1=u_2v_2\)

These 2 equations have infinitely many integer solutions.

Is there a way to prove this?

 

[tkhunny]

Can you use two to construct a third?

 

[procyon]

Can you use two to construct a third?

Yes

I assume you are thinking of then solving by elimination?

If so, I don't believe a 'new' equation, created from the first two, will have any meaningful use as it is, by definition, dependant on the first two.

i.e. you could do this

\(\displaystyle (u_1+v_1)^2 = (u_2-v_2)^2 ....(1)\)

\(\displaystyle 4u_1v_1=4u_2v_2 ......................(2)\)

This 'appears' useful because you then have

\(\displaystyle (1) + (2) \implies (u_2+v_2)^2-(u_1+v_1)^2=4u_1v_1\)

If you continue on with solving this you will end up with fractional expressions that are in the same form as the original two. So no progress will have been made.

 

[tkhunny]

No, not a new equation, a new solution.

If you can build them, and it's not cyclic, you have infinitely many solutions.

 

[procyon]

No, not a new equation, a new solution.

If you can build them, and it's not cyclic, you have infinitely many solutions.

Not possible me thinks.

They are part of 4 equations where I'm trying to perform a negative proof.

I don't want to give up yet, so I'll wait to post the hypothesis for another while , or hopefully the proof