Solving A Cool Diophantine Equation With Integers
VloĆŸit
- Äas pĆidĂĄn 6. 07. 2024
- đ€© Hello everyone, I'm very excited to bring you a new channel (SyberMath Shorts).
Enjoy...and thank you for your support!!! đ§Ąđ„°đđ„łđ§Ą
/ @sybermath
/ @aplusbi
â Join this channel to get access to perks:â bit.ly/3cBgfR1
My merch â teespring.com/stores/sybermat...
Follow me â / sybermath
Subscribe â czcams.com/users/SyberMath?sub...
â Suggest â forms.gle/A5bGhTyZqYw937W58
If you need to post a picture of your solution or idea:
intent/tweet?text...
#numbertheory #diophantineequations #integers
via @CZcams @Apple @Desmos @NotabilityApp @googledocs @canva
PLAYLISTS đ” :
ⶠTrigonometry: ⹠Trigonometry
ⶠAlgebra: ⹠Algebra
ⶠComplex Numbers: ⹠Complex Numbers
ⶠCalculus: ⹠Calculus
ⶠGeometry: ⹠Geometry
ⶠSequences And Series: ⹠Sequences And Series
The unsaid fact: n(n+1)/2 is the nth triangular number. Also, any time 8n+1 (where n is any variable or expression representing an integer) turns up alone under a square root, you can be sure that triangular numbers are lurking somewhere. A bit like if n^2-n-1 shows up then there's usually a golden ratio afoot.
If x + y = k^2 then x - y can be k or (-k). Hence the 2 solutions you got with the 1st method.
took me a while of bashing my skull in to the wall with congruence, parity, perfect squares and divisibility, but then the solution hit me and i felt kind of dumb..
Set x-y=n, therefore x=y+n
Replace in our original equation to get n+y+y=n^2
Cleaning up you get y = œ(n^2 - n)
Do the same for y=x-n, x= œ(n^2 + n)
The solution set to our original equation is (x, y) = [œ(n^2 + n), œ(n^2 - n)] for every integer n
That type of struggle makes us better. Nice! đ
x=1 and y=0 also works
For the 2nd method, to avoid invalid solution value w/o checking, note that
x+y=(x-y)ÂČ --> x+y is positive
It means that
âą both x and y can't be negative
âą both x and y are positive or
âą one of x and y is positive and the other one is negative but the absolute value of the negative one is less than the positive. To be clearer, let say y
*x and y are both equal to 0*
only for (0, 0)?
X= 0, Y=0.
In #2 k and -k give (x,y) and (y,x)