Discrete maths proof [HELP PLEASE]

D1Kuta

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Prove there are no positive integer solutions to x^2+x+1=y^2 by contradiction.

All I got was that up to was realizing

x^2+x=y^2-1
x(x+1)=(y-1)(y+1)

but I got stuck at this bit, not sure what to do next and how to get the solution... can anyone help please?:(
 
Prove there are no positive integer solutions to x^2+x+1=y^2 by contradiction.
If there were a positive integer solution x\displaystyle x then y\displaystyle y must also be an integer.
Thus you get
x2+x+(1y2)=0\displaystyle x^2+x+(1-y^2)=0
Does that mean that 14(1)(1y2)\displaystyle \sqrt{1-4(1)(1-y^2)} has to be a even integer?
 
Last edited:
Prove there are no positive integer solutions to x^2+x+1=y^2 by contradiction.

All I got was that up to was realizing

x^2+x=y^2-1
x(x+1)=(y-1)(y+1)

but I got stuck at this bit, not sure what to do next and how to get the solution... can anyone help please?:(
How about x(x+1) must be odd while (y-1)(y+1) must be even? Why is this true?

How can you make up a contradiction proof using this?
 
How about x(x+1) must be odd while (y-1)(y+1) must be even? Why is this true?
If x=2\displaystyle x=2 then x(x+1)\displaystyle x(x+1) is EVEN not ODD .
If x=3\displaystyle x=3 then x(x+1)\displaystyle x(x+1) is EVEN not ODD
If y=2\displaystyle y=2 then (y1)(y+1)\displaystyle (y-1)(y+1) is ODD not EVEN.
If y=3\displaystyle y=3 then (y1)(y+1)\displaystyle (y-1)(y+1) is EVEN not ODD .
 
If x=2\displaystyle x=2 then x(x+1)\displaystyle x(x+1) is EVEN not ODD .
If x=3\displaystyle x=3 then x(x+1)\displaystyle x(x+1) is EVEN not ODD
If y=2\displaystyle y=2 then (y1)(y+1)\displaystyle (y-1)(y+1) is ODD not EVEN.
If y=3\displaystyle y=3 then (y1)(y+1)\displaystyle (y-1)(y+1) is EVEN not ODD .
I did not so any of this. Someone else posted it. I will never admit that I wrote this!
 
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