# Diophantine Equation With Varying Exponents

I am considering the following Diophantine Equation - the approach I tried became the study of too many different cases - so many that I left it and tried to find an easier way. I wonder if anyone could shed any further light on the matter. The equation is interesting since it has 3 different exponents. $$a^2 +b^3 = c^4$$ $a$, $b$, $c$, are any integers. The approach I mentioned began by subtracting the $a^2$ on both sides and factoring the difference of two squares - but as I said, it quickly became very tedious.

• What are we seeking? A solution where none of the variables is $0$? Infinitely many solutions of this type? All solutions? There are other possibilities. Jan 10, 2015 at 22:17
• All solutions. The case when one of the variables is 0 included, although these are trivial and easy to deduce. Jan 10, 2015 at 22:23
• see Lemma 3.2.7 p.22 here cecm.sfu.ca/~nbruin/thesis.pdf . Just set $(a,b,c)\rightarrow (a,-b,c)$ in the set of solutions.
– 111
Jan 10, 2015 at 22:24
• Really, so one can obtain a non-trivial solution from those parametric solutions? Jan 10, 2015 at 22:26
• I needed it,in my phd...
– 111
Jan 10, 2015 at 22:48

Here's an infinite family of solutions,

$$\big(4q^2(p^2-2)\big)^2+(4dq^2)^3 = (2pq)^4$$

where $p,q$ solve the Pell equation $p^2-d^3q^2=1$. If you want the "reversed" form,

$$p^4 + (q^2-1)^3 = (q^3+q)^2$$

where $p^2-3q^2=1$. And an amazing one by Enrico Jabara,

$$(2^{88}7^{47}n^4)^3 + (2^{92}7^{49}n)^4 + (2^{32}7^{17}n^4)^5 + (2^8 7^4n^4)^7 + (n^4-2^{52}7^{28})^8 = (n^4+2^{52}7^{28})^8$$

which he probably found by expanding,

$$(an^4)^3 + (bn)^4 + (cn^4)^5 + (dn^4)^7 + (n^4-e)^8 = (n^4+e)^8$$

collecting powers of $n$, and solving for $a,b,c,d,e$.

I thought it possible to write a solution without using equations Pell.

$$a^2+b^3=c^4$$

If you make this change.

$$p=tz(2zk^2+t)$$

$$s=tzk^2(2zk^2-t)$$

The result of such decision.

$$a=sp^3$$

$$b=2tzk^2p^2$$

$$c=kp^2$$

Where the number $t,z,k$ - integers and set us. You may need after you get the numbers, divided by the common divisor.