Interesting numbers

Hi everyone, an "interesting number" is defined as x^2 + y^3 , with x,y both integers.

How can I prove that number 6 is/isn´t an "interesting number"?

x^2 + y^3 = 6 ? Find values for x and y, both integers

Thanks for trying!

#NumberTheory

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

I looked at integral points on the elliptic curve $Y^2 = X^3 + D$ to get a list of interesting numbers $\le 100.$

$6$ is not interesting. Neither is $7,11, 13, 14, 20, \ldots.$ There are 41 uninteresting positive integers $\le 100.$

Some interesting small interesting numbers are $\begin{aligned} 18 &= 19^2-7^3 \\ 19 &= 12^2-5^3 \\ 30 &= 83^2-19^3 \\ 38 &= 37^2-11^3 \\ 79 &= 302^2-45^3 \\ 97 &= 77^2 - 18^3 \\ \end{aligned}$

Here is a link to the OEIS entry for this sequence.

Patrick Corn - 2 years ago

Thanks for this helpful answer!

Iñaki Salido - 2 years ago

Mathematic is my favorite, but hardest subject. I study it in my college a lot, but it's very hard.

Veronica Pauli - 1 year, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$