still spear
Q: find all nonzero integral m such that m^2-8m is a perfect square
Work: tried factoring and completing the square, couldnt find anywhere to go
I put this in the university section because this is not high school math (at least in the USA), but I have NOT taken a college course on number theory (just familiar with basic mod techniques and theorems, i've done AOPS Intro to Number Theory and AOPS Intermediate Algebra)
crimson phoenixBOT
Q: find all nonzero integral m such that m^2-8m is a perfect square
Work: tried ...
+close
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still spear
1. **Do not ping the Moderators**, unless someone is breaking the rules.
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s h u t
ruby cargo
Q: find all nonzero integral m such that m^2-8m is a perfect square
Work: tried ...
why didn't completing the square work?
still spear
why didn't completing the square work?
didnt know where to go from that
ruby cargo
Here's a hint :
squares grow farther and farther apart as the general values increase
still spear
(x-4)^2 - 16
ruby cargo
nice!
still spear
but 16 is negative so i cant rly relate this to any pythagorean triple or smth
ruby cargo
(x-4)^2 - 16
so we have a square that is 16-away from another square
ruby cargo
there are only so many of those, prove this and proceed
still spear
but 16 is negative so i cant rly relate this to any pythagorean triple or smth
i can 😮
ruby cargo
there are no Pythagorean triples involved
ruby cargo
use the fact that if $a, b$ are distinct squares such that $a, b>N$ then $|a-b|>f(N)$ for some strictly increasing function $N$
dense fjordBOT
Coffey
ruby cargo
use the fact that if $a, b$ are distinct squares such that $a, b>N$ then $|a-b|>...
you don't need to write all this, I'm just trying my best to not give you the answer
ruby cargo
if (x-4)^2 = k^2 +16
you might have found most solutions but you have to prove that there are finitely many too
still spear
you might have found most solutions but you have to prove that there are finitel...
oh right
oh theres also x^2 - 8x = 0 so x=8
how do i prove (-1, 8, 9) are the only solutions tho
ruby cargo
by reading what I said above
still spear
if (x-4)^2 = k^2 +16
can i just say that this is a quadratic in x and has 2 solutions
still spear
oh theres also x^2 - 8x = 0 so x=8
and this is a special case
still spear
**Coffey**
ohh wait because this is strictly increasing there cant be any more after
thx
+close
rough sinewBOT
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