I just figured this out, maybe some of you already have. Anyway, I was trying to find easier ways to memorize the notes on the fretboard and I started looking at notes adjacent to eachother and I hit on this mnemonic.

I've since seen it mentioned in relation to the circle of fifths, but I'm not too strong on that subject. Anyway, the mnemonic is BEADGCF. This works everywhere, except between the G and B strings where the symmetry is broken. But basically it means that if you have a B on a lower string, say fret 7 on the low E, then you'll have an E directly below. And directly below that E an A and below that a D and so on. So below the C on the 8th fret E string is an F. I just found it really helped me map things out.

Thought it might help someone else out.
That makes sense, because if you think of a basic (two string) powerchord, that's your root note and then the fifth. The relationship you mentioned is the root note and the fourth, which is one whole step down from the fifth.
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Good way of looking at it though its not like a cycle since below F would be B flat then below B flat would be ...etc etc. But as far as those notes go its good.
i remembered all the fretboard by only remembering the 1st string, the 5th string and the 6th string....and maybe a little of the 2nd string..
I use octave logics. Octave comes like over one string and 2 frets (or 3 at some case) forward.
so if i'd want to know say string 4, fret 7...then i look at string 6 and fret 5...
i know that the the 7th frets on the 3rd-6th strings spell 'BEAD'. Also if u use 3 notes for a power chord, the lowest and highest (1st and 3rd) note in the chord is the same letter(with different pitch).
fretboard warrior is a good wee program to help you memorise the notes on the fretboard.
It uses a story to help you memorise all the notes.