How many solutions?

Number Theory Level 3

$\large \alpha^2 + \beta^2 = 4^\gamma$

Find the number of positive integer triplets $(\alpha, \beta, \gamma)$ that satisfy the equation above.

1 There is no solution There are infinitely many solutions 49

1 solution

Bloons Qoth
Jun 19, 2016

${x^2+y^2=4^z}$
${x^2+y^2=4^1}$ ..........................There are no natural solutions satisfying $x$ and $y$
${x^2+y^2=4^z}$ where $z>1$
${(x^2+y^2) mod 8=4^z mod 8}$
2 cases: even or odd
.....If $x$ and $y$ are odd, $1+1=0$ ?
.....If $x$ and $y$ are even, $0+0=0$ ?.............................The even case makes more sense.
Therefore, $x$ and $y$ are even.
$z \rightarrow z+1$ ............................If $z$ works for the next number
${x^2+y^2=4^{z+1}}$
If $x=2n$ and $y=2k$
${(2n)^2+(2k)^2=4^{z+1}}$
${4n^2+4k^2=4^{z+1}}$
${n^2+k^2=4^z}$
Proof by infinite descendant/induction. Therefore, no natural solution exists for $(x,y,z) \blacksquare$