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Mr.Math
 3 years ago
This group hasn't been active lately. So I'm going to post a set of fun problems bring it back to life.
\[\text{ Fun With Mr.Math #1}\]
Find the integer solutions to the equation
\[9^x3^x=y^4+2y^3+y^2+2y.\]
Mr.Math
 3 years ago
This group hasn't been active lately. So I'm going to post a set of fun problems bring it back to life. \[\text{ Fun With Mr.Math #1}\] Find the integer solutions to the equation \[9^x3^x=y^4+2y^3+y^2+2y.\]

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Tomas.A
 3 years ago
Best ResponseYou've already chosen the best response.0why wolframalpha doesn't give solution as x=0 and y=0? http://www.wolframalpha.com/input/?i=9%5Ex3%5Ex%3Dy%5E4%2B2y%5E3%2By%5E2%2B2y+solve+integer

Mr.Math
 3 years ago
Best ResponseYou've already chosen the best response.3I don't know. I wish it doesn't give any solution at all :D

sasogeek
 3 years ago
Best ResponseYou've already chosen the best response.0if x=0, y=an imaginary number

Tomas.A
 3 years ago
Best ResponseYou've already chosen the best response.09^03^0=11=0 and when y=0 then 0+0+0+0=0 same answer, no?

sasogeek
 3 years ago
Best ResponseYou've already chosen the best response.0factor out y, you'll end up with \(\ (y^2+2y)(y^2+1)=9^x3^x \) when x=0, \(\ y^2=1, y=\sqrt{1} \)

sasogeek
 3 years ago
Best ResponseYou've already chosen the best response.0but \(\ y^2+2y=0 => y=2 \)

Tomas.A
 3 years ago
Best ResponseYou've already chosen the best response.0y(y+2)(y^2+1)=0 so y=0 and y=2 no?

sasogeek
 3 years ago
Best ResponseYou've already chosen the best response.0please write full statements lol, you're getting me lost by ... no? :P

Tomas.A
 3 years ago
Best ResponseYou've already chosen the best response.0you write \[(y^2+2y)(y^2+1)=9^x3^x\] if x=0 \[(y^2+2y)(y^2+1)=0\] \[y(y+2)(y^2+1)=0\] so \[y=0\text{ and }y+2=0\text{ and }y^2+1=0\] so y=0 y=2 and no real solutions

sasogeek
 3 years ago
Best ResponseYou've already chosen the best response.0=> my solutions is wrong? not sure... i guess i'll wait for to see the correct one :D

Tomas.A
 3 years ago
Best ResponseYou've already chosen the best response.0i don't care about solution because i dont have a clue how to solve but x=0 and y=0 is clearly true and u didn't prove me that it's wrong lol

sasogeek
 3 years ago
Best ResponseYou've already chosen the best response.0:P y=2 and y=0 both work for x=0 and are true, but i wonder if they're the right solutions lol, Mr. Math, what are the right solutions?

Tomas.A
 3 years ago
Best ResponseYou've already chosen the best response.0so why wolframalpha does not show y=0?

sasogeek
 3 years ago
Best ResponseYou've already chosen the best response.0it isn't as smart as u ;)

asnaseer
 3 years ago
Best ResponseYou've already chosen the best response.3Here are my thoughts so far: First lets simplify the left hand side:\[9^x3^x=(3^2)^x3^x=3^{2x}3^x=3^x(3^x1)\]Then lets simplify the right hand side:\[y^4+2y^3+y^2+2y=y(y+2)(y^2+1)\] We therefore have:\[3^x(3^x1)=y(y+2)(y^2+1)\]Now the trivial solution found above is x=0 which leads to y=0 or y=2 (since we are only interested in integer solutions). Another solution could also be \(x=\infty\) and y=0 or 2 but I assume we can rule out infinities. Now lets rearrange the equation to:\[y^2+1=\frac{3^x(3^x1)}{y(y+2)}\]This means \(y(y+2)\) must evenly divide into \(3^x(3^x1)\). Notice that \(3^x\) is always an odd number and \(3^x1\) is always an even number. Case 1: If y is an even number, then y+2 will also be even. Therefore, if y is even, then \(y(y+2)\) must evenly divide into \(3^x1\). So let \(y=2m\) to get:\[\frac{3^x1}{4m(m+1)}\hspace{2cm}\text{must be an integer}\] Case 2: If y is an odd number, then y+2 will also be odd. Therefore, if y is odd, then \(y(y+2)\) must evenly divide into \(3^x\). So let \(y=2m+1\) to get:\[\frac{3^x}{(2m+1)(2m+3)}\hspace{2cm}\text{must be an integer}\]

asnaseer
 3 years ago
Best ResponseYou've already chosen the best response.3Since 3 is a prime number, case 2 can be rejected because \((2m+1)(2m+3)\) can only evenly divide into \(3^x\) if \((2m+1)(2m+3)=3^k\) which is impossible. So we are left with just Case 1 to solve.

asnaseer
 3 years ago
Best ResponseYou've already chosen the best response.3In case 1, if m>1, then \(4m(m+1)\) will always contain a product of an even number and an odd number. However, we know the numerator \(3^x1\) is always even, therefore we can reject all values of m>1. So case 1 reduces to m=1 which gives y=2 (since \(y=2m\) for this case). Substituting this back into the original equation gives:\[\begin{align} 3^x(3^x1)&=y(y+2)(y^2+1)\\ &=2(2+2)(2^2+1)\\ &=2*4*5\\ &=40 \end{align}\]which does not give an integer solution for x. Therefore (I believe) the only solutions are: 1) x=0 and y=0 or 2 2) x=\(\infty\) and y=0 or 2

Mr.Math
 3 years ago
Best ResponseYou've already chosen the best response.3Great job everyone, some solution(s) is/are still missing though.

asnaseer
 3 years ago
Best ResponseYou've already chosen the best response.3hmmm  I rechecked my work and found some flaws in my logic. I have now proved that for y>0 y must be odd. And, by trialanderror I have found one more solution: 3) x=1 and y=1 but I have not been able to prove that this is the only other solution :(

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.0its immediate that \(x\ge0\) letting \(x=0\) gives 2 solution \[(x,y)=(0,0)\ , \ (0,2)\]suppose that \(x>0\) using asnaseer factoring\[3^x(3^x1)=y(y+2)(1+y^2)\]RHS is divisible by \(3^x\)\[y \equiv 0,1,2 \ \ \text{mod} \ 3 \\ y^2 \equiv 0,1 \ \ \text{mod} \ 3 \\ 1+y^2 \equiv 1,2 \ \ \text{mod} \ 3 \]so \(y(y+2)\) is divisible by \(3^x\) but we know that \(\gcd(y,y+2)=1,2\) so exactly one of \(y\) or \(y+2\) is divisible by \(3^x\). case 1 : \(y=3^xm\) and \(m\neq0\) and \(m\) is not a multiple of 3\[3^x1=m(3^xm+2)(1+m^23^{2x})\]RHS is greater than LHS in magnitude so no solution from here case 2 : \(y+2=3^xm\) and \(m\neq0\) and \(m\) is not a multiple of 3\[3^x1=(3^xm2)m(5+m^23^{2x}4m3^x)\]\[3^x1=m(3^xm2)(5+(3^xm^24m)3^x)\]again RHS is greater than LHS in magnitude unless \(x=1\) so one solution from here\[(x,y)=(1,1)\]

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.0actually for case 2 if \(x>1\)\[m(3^xm2)(5+(3^xm^24m)3^x)>(5+(3^xm^24m)3^x)>5+3^x\]

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.0only solutions\[(x,y)=(0,0)\ ,\ (0,2)\ , \ (1,1)\]
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