Let's define a sandwich number such that it is sandwiched between a perfect square and a perfect cube. For example, 26. $26-1=5^2$ and $26+1=3^3$. We can use the equation $y^3-x^2=2$ to define these sandwich numbers. We will assume the cube is larger than the square for this problem. (I was instructed to use quadratic integers to solve this problem.)
- Show that if $y^3-x^2=2$ then $y$ must be odd and from that show that $x$ must be odd.
I have attempted this by looking at the equation $\bmod 8$.
What are the units in this quadratic field, and is $2$ a prime in this quadratic field?
If x is any odd standard rational integer what can be said about the greatest common factor of the quadratic integers $(x+\sqrt{-2})$ and $(x-\sqrt{-2})$ in Q[$\sqrt{-2}$]?
If $\alpha$ and $\beta$ are relatively prime quadratic integers in Q[$\sqrt{-2}$] and their product is a perfect cube, must $\alpha$ and $\beta$ themselves be perfect cubes?
If $x$ is any odd standard rational integer and $a$ and $b$ are standard rational integers for which:
$x+\sqrt{-2}$=$({a+b\sqrt{-2}})^3$
How can I use binomial expansion to solve for $x$?
Lots of questions, but there are some positive answers: units are only $-1$ and $1$, $2=-\sqrt{-2}\cdot\sqrt{-2}$ is not a prime, there's an Euklidean algorithm to find the greatest common divisor, and thus factorization into prime elements is essentially unique (meaning the answer to 4. is "yes"). The answer to 5. is obvious, just expand it, and compare both parts (both the coefficients of $\sqrt{-2}$ and the rest). BTW, references for the facts I mentioned can be found here: https://en.wikipedia.org/wiki/Quadratic_field#Orders_of_quadratic_number_fields_of_small_discriminant