Methods for quartic diophantine equation

$$3x^2 + 2y^4 = z^4$$

How do I solve this?? I would like to use so-called "elementary number theory", not abstract algebra (e.g. $\mathbb{Z} ( \sqrt d)$) or elliptic curves.

Note: I'm not asking what the solutions are, but rather how to find them.

My instincts are:

• search the internet (I compared this equation with the ~280 here on MSE, and tried a variety of similar searches on uniquation.com ...)
• search the 3 number theory books that I have
• try to find solutions "by inspection" (possibly after reducing the order of the variables)
• do some magic with modular arithmetic
• use Alpern's solver - which seemed to indicate that there are no solutions (though I might have made an illegal substitution, so to speak)

I was able to identify $A = 6, B = 3, C = 6$ as solutions of $ \ 3A + 2B \ ^2 = C \ ^2$, but those aren't squares!

What is the number-theoretic approach to such problems? Is there a general method?

Solution 1:

Supposing we did have a solution lets consider the equation modulo $3$, since a square (hence a fourth power) must be congruent to $0$ or $1$ so the LHS is congruent to $0$ or $2$ and the RHS is $0$ or $1$ we see $3$ must divide both $y$ and $z$ thus $3^3$ must divide $x^2$ so $3^2$ divides $x$ hence $3^4$ divides the entire equation and dividing through leaves the same equation as we started with so we fall into an infinite decent which is absurd, hence we can have no solution in integers.

Solution 2:

Maybe what you need is Legendre's Theorem. Certainly it covers this situation. It tells you exactly what needs to be checked. It is presented in Ireland and Rosen, A Classical Introduction to Modern Number Theory, chapter 17, section 3. A very similar treatment is in PETE, pages 5-8.

Anyway, if $a,b,c$ are integers, not all positive and not all negative, but all nonzero, they are all squarefree, $\gcd(b,c) = \gcd(c,a) = \gcd(a,b) = 1,$, then $$ a x^2 + b y^2 + c z^2 = 0 $$ has a nontrivial solution in integers if and only if all three of these are true:

(i) $-bc$ is a square $\pmod a,$

(ii) $-ca$ is a square $\pmod b,$

(iii) $-ab$ is a square $\pmod c.$

In these we include $0$ as a square.

This result is also done in any book on quadratic forms. I like Rational Quadratic Forms by J. W. S. Cassels.

Anyhoo, you have $$ 3 U^2 + 2 V^2 - W^2 = 0,$$ so $a=3, \; b=2, \; c = -1.$ Condition (iii) asks if $-6$ is a square $\pmod {-1},$ the answer is yes as it is a multiple. Condition (ii), is $3$ a square $\pmod 2,$ again the answer is yes. However, condition (i), $2$ is not a square $\pmod 3.$ So there you are.

Leonard Eugene Dickson devoted a whole chapter to this in Introduction to the Theory of Numbers (1929). He still gave a section on it in Modern Elementary Theory of Numbers (1939).

NOTE: so this problem can be done without the fourth powers. The traditional one where the fourth powers really matter is $X^4 + Y^4 = Z^2,$ which is part of the proof of Fermat's Last Theorem, and is not just an application of Legendre's Theorem (although not difficult).