Interesting cubes

I got this while solving this Problem: Three cubes and a square

I was working for the answer (laboriously). I then found one special cube number.

It was 216. That number can be expressed as the sum of three consecutive cubes;27, 64 and 125 surprisingly add up to the next cubic number.

I also saw that cube roots of the numbers are in the Pythagorean Triples (3,4,5). However, Pythagorean triples are self-contained but these cubes have the sum of an external cube. The question is : Is there another number that has this property?

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

There is a formula by Ramanujan to generate cubes that are the sum of three other cubes.

$(3a^2+5ab-5b^2)^3 + (4a^2-4ab+6b^2)^3 + (5a^2-5ab-3b^2)^3 = (6a^2-4ab+4b^2)^3$

I found it here.

There's also

$27^5 + 84^5 + 110^5 + 133^5 = 144^5 \text{(Lander \& Parkin, 1966)}$

$(-220)^5+ 5027^5 + 6237^5 + 14068^5 = 14132^5 \text{(Scher \& Seidl, 1996)}$

$55^5 + 3183^5 + 28969^5 + 85282^5 = 85359^5 \text{(Frye, 2004)}$

From here

Henry U - 2 years, 7 months ago

Thank you, Henry

Mohammad Farhat - 2 years, 7 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}`	$\boxed{123}$