Terence tao's list of mathematical embarrassments

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A mathematical embarrassment (ME) is a problem that should have been solved by now. An ME usually is easy to state, seems approachable, and yet resists all attempts at an attack. There may be many reasons that they are yet to be solved, but they “feel” like they should have been solved already.
i didn't think the invariant subspace problem would be an embarrassment. i guess it's not all that hard for finite dimensions. only one person mentioned goldbach's conjecture, & nobody said anything about perfect numbers. they seem pretty easy to understand but hard to say interesting things about. same with fermat's last theorem & the 4-colour problem
 
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I have recently read that someone figured out how to write ##33## as a sum of three cubes and that now ##42## is the next number to target.

The cubes have been so absurd, that I wouldn't call it an embarrassment although it fits the definition above.
##(8,866,128,975,287,528)^3 + ( - 8,778,405,442,862,239)^3 + (-2,736,111,468,807,040)^3 = 33##
 
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i didn't think the invariant subspace problem would be an embarrassment. i guess it's not all that hard for finite dimensions. only one person mentioned goldbach's conjecture, & nobody said anything about perfect numbers. they seem pretty easy to understand but hard to say interesting things about. same with fermat's last theorem & the 4-colour problem
As to perfect numbers, we know all the even ones, because every Mersenne prime has a corresponding perfect number, and such perfect numbers are all the even ones. However, we have been unable to find any odd perfect numbers, and we have been unable to prove that there are none.

As to Mersenne numbers ##M_p = 2^p-1## for primes ##p##, we don't know if a finite number or infinite number of them is prime or else is composite. Likewise, of the Fermat numbers ##F_n = 2^{2^n}+1##, the lowest five are prime, and all other ones that we have been able to test are composite. We don't know if there are any other Fermat primes, or else if the first five are all of them.

Terence Tao himself has found partial solutions to Goldbach's conjecture: Terence Tao releases partial solution to the Goldbach conjecture « Math Drudge, Goldbach conjecture | What's new

As to Fermat's Last Thorem and the Four-Color Hypothesis, the problem is that their proofs are *very* complicated.
 

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