Question: I've been having this doubt for a quite a while and haven't been able to find a satisfying answer. Is the 'E#' note technically an F? If so, why does it not exist on the chromatic scale like other sharps? I mean, what exactly is the E#? Is it a note on its own or is it just another way of saying F?

And what do people mean when they say that the D# Major scale is theoretical major scale and why it does not exist in the circle of fifths diagram?

I came across the last paragraph on a website.

Answer: 
1. E-sharp is called the enharmonic equivalent of F-natural. The sharp raises the base note one half step. In the same way B-sharp is the enharmonic respelling of C-natural. Furthermore, C-flat is the enharmonic of B-natural, and F-flat enharmonic of E-natural. Notes may also be double-sharped and double-flatted, which raises the base note by two half steps.
2. The key of D-sharp could be considered a theoretical scale because it doesn’t appear on the basic circle of fifths is simply because it is a scale that contains these enharmonics, and traditionally composers write this as E-flat to avoid the enharmonic spelling of the second scale degree, F-natural, which is enharmonic with E-sharp, the second degree of D-sharp major, and no one likes reading parts with double-sharps in the key signature (shudder). The seventh degree of the D-sharp major scale is C-double-sharp. D-sharp major chords definitely exist in music and we play enharmonic spellings all the time.
(In addition to Tim's great answer, I wanted to introduce another important term and a slightly different way of thinking of it.)

Whenever we see a pitch that is raised from its expected form in the diatonic scale, it is often (although not always) understood as a *temporary leading tone*. By "leading tone" we mean it's the seventh scale degree, and since E♯ is the seventh scale degree of an F♯ scale, we say that this E♯ is the temporary leading tone of F♯.

Even though this E♯ is the exact same note as F on a modern equally tempered piano, their spellings tell us how the two pitches function. E♯ is scale-degree 7 of F♯, whereas F is scale-degree 7 of G♭. Even though E♯ and F sound the same—we say that they are \*enharmonic—they have very different functions.

As for D♯ major being a theoretical key, this just means that it has at least one doublesharp or doubleflat in its key signature. D♯ actually has *two* doublesharps—F and C—so it is a theoretical key. (See also [Where do the double accidentals go in "theoretical" key signatures?](https://music.stackexchange.com/questions/65936/where-do-the-double-accidentals-go-in-theoretical-key-signatures))