Dtjytyk

It is well known that there exist integer solutions to the equation $a^2+b^2=c^2$.
For example, an explicit formula for integer values of $a$ , $b$ , and $c$ is

begin{align}a&=2mn \ b&=m^2-n^2 \ c&=m^2+n^2.end{align}

I want to find solutions to the equation $a^2+b^2=c^4$. However, I got $a,b,c$ in a general form that substitutes $x$ in integer and get $a,b,c$ in the integer. What should I do first? I'm very stuck!

Your equation, $a^2 + b^2 = c^4$, is the same as $a^2 + b^2 = c^2$, except that $c$ has been replaced with $c^2$. Thus you can find solutions by choosing $m$ and $n$ so that
$$
begin{align}
a&=2mn \ b&=m^2-n^2 \ c^2&=m^2+n^2.
end{align}
$$
But how can you make sure that $c$ is an integer? Easy: that last equation, $c^2 = m^2 + n^2$ says that $m,n$ and $c$ also form a Pythagorean triple. Thus you can find solutions with two new parameters, $r$ and $s$ such that
$$
begin{align}
m&=2rs \ n&=r^2-s^2 \ c&=r^2+s^2.
end{align}
$$
Substituting these in the first set of equations to find $a$ and $b$, you have
$$
begin{align}
a&=4rs(r^2-s^2) \ b&=6r^2 s^2-r^4-s^4 \ c&=r^2 + s^2.
end{align}
$$
For example, choosing $r=2$ and $s=1$ gives you $a = 24$, $b=7$, $c=5$ and, indeed, $24^2 + 7^2 = 5^4$.

Whenever you have $a^2+b^2=c^2$ you also get $(ca)^2+(cb)^2=c^4$ -- does that count for you?

You are looking for a Pythagorean triple where the hypotenuse is an odd square. You can find if such a triple exists if you try values of $n$ in the following formula where $C$ is the odd square: $$k=frac{-(2n-1)+sqrt{2C-(2n-1)^2}}{2}$$
from $1$ up to the point where $n^2+1>C$. If $k$ is an integer for any value(s) of $n$, then you can find A,B,C substituting $n$ and $k$ into the following functions:
$$A=(2n-1)^2+2(2n-1)k$$
$$B=2(2n-1)k+2 k^2$$
$$C=(2n-1)^2+2(2n-1)k+2k^2$$

If no integer $k$ is found for any value tested up to where $n^2+1>C$, then no triangle exists for that value of C. Remember that C must be an odd square for this to work because you are looking for one where $C^2=c^4$.

Some other triples that satisfy this equation are (41, 840, 29),
(239, 28560, 169),
(63, 216, 15),
(369, 7560, 87),
(175, 600, 25),
(1025, 21000, 145),
(161, 240, 17),
(343, 1176, 35),
(721, 5280, 73),
(959, 9360, 97),
(567, 1944, 45),
(1183, 4056, 65),
(1575, 5400, 75),
(527, 336, 25),
(1241, 2520, 53),
(2023, 6936, 85),
(3281, 18480, 137),
(2527, 8664, 95),
(1071, 1080, 39),
(1449, 2160, 51),
(3087, 10584, 105),
(1081, 840, 37),
(2047, 3696, 65),
(3703, 12696, 115),
(4375, 15000, 125),
(5103, 17496, 135),
(5887, 20184, 145),
(1519, 720, 41),
(4681, 10920, 109),
(6727, 23064, 155),
(7623, 26136, 165),
(2975, 3000, 65),
(4025, 6000, 85),
(8575, 29400, 175),
(9583, 32856, 185).