Math Trivia -- A sum of cube roots of certain consecutive positive integers

[lookagain]

There is nothing here for which to solve. I was experimenting, and I discovered that the following sum
is relatively close to an integer:

\(\displaystyle \sqrt[3]{2} \ + \ \sqrt[3]{3} \ + \ \sqrt[3]{4} \ + \ \sqrt[3]{5} \ \approx \ 6 \)


The sum is approximately equal to 5.9995476.

 

[Dr.Peterson]

There is nothing here for which to solve. I was experimenting, and I discovered that the following sum
is relatively close to an integer:

\(\displaystyle \sqrt[3]{2} \ + \ \sqrt[3]{3} \ + \ \sqrt[3]{4} \ + \ \sqrt[3]{5} \ \approx \ 6 \)


The sum is approximately equal to 5.9995476.

For more similar facts, see

Almost Integer -- from Wolfram MathWorld

An almost integer is a number that is very close to an integer. Near-solutions to Fermat's last theorem provide a number of high-profile almost integers. In the season 7, episode 6 ("Treehouse of Horror VI") segment entitled Homer^3 of the animated televsion program The Simpsons, the equation...

mathworld.wolfram.com

Mathematical coincidence - Wikipedia

en.wikipedia.org

I don't find yours there (though I didn't scan closely). The important thing is, people do find these interesting!

 

[BigBeachBanana]

This reminds me of this question (which can be solved algebraically).
[math]\sqrt[3]{x} + \sqrt[3]{x+1}+ \sqrt[3]{x+2} = 6[/math]
There are 1 real, and 2 complex roots.

 

[BigBeachBanana]

For more similar facts, see

Almost Integer -- from Wolfram MathWorld

An almost integer is a number that is very close to an integer. Near-solutions to Fermat's last theorem provide a number of high-profile almost integers. In the season 7, episode 6 ("Treehouse of Horror VI") segment entitled Homer^3 of the animated televsion program The Simpsons, the equation...

mathworld.wolfram.com

Mathematical coincidence - Wikipedia

en.wikipedia.org

I don't find yours there (though I didn't scan closely). The important thing is, people do find these interesting!

The one relating pi and the golden ratio is still a prominent on-going conspiracy.