Alan

"I assume prime triplets are any 3 consecutively prime numbers ....    Is that correct?    For now I will assume so."

"3 consecutive prime numbers" is a necessary, but not sufficient condition, Melody.  For example: 31, 37 and 41 are consecutive primes, but they don't have the form (p, p+2, p+6) or (p, p+4, p+6) that define prime triplets (as GingerAle pointed out above).

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PPMCC:  Pearson Product Moment Correlation Coefficient?

Don 't know of a simple way by hand other than slogging through the formula!

However, you could use the online calculator at: http://www.socscistatistics.com/tests/pearson/Default2.aspx 

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"Then how do you know so much about things and stuff?"  

I don't know very much about anything!  I've applied math to real world systems for a long time though.

"What university did you graduate from?"

Warwick University  (in the UK)

Unoccupied volume = 3709.44 - 2400 = 1309.44 cm³

As a percentage of the total volume of the box this is: (1309.44/3709.44)*100  ≈ 35.30%

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I'm afraid my knowledge is insufficient here. Perhaps this will help:

http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/tracenorm.pdf

I am not and never have been a professor!

Did I fail you last time?!!!

I haven't checked, and it's a long time since I did this sort of thing, but:

1. I think it wouldn't make sense to use anything other than the field's own definition of addition in calculating the trace (sum of diagonal elements).

2. Hence the trace must be a member of the field.

If the answer is important then you should double check this with someone more qualified to answer!

Something like:

xav_new = xav_old + (x_i+1 - x_i-5)/5

This assumes your first xav is based on the first five values of x, and that i is an index that points to the 5th value of the current value of xav, just prior to calculating the next value.

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Use the formulary here:  https://web2.0calc.com/formulary/math/geometry/cylinder 

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You need a graphical or numerical method to find the value of x here.

It's a good idea to draw a graph anyway to get a good initial estimate for an iterative numerical method:

So x ≈ -0.204

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??!!     Any number mod 24 is going to lie in the range 0 to 23    !!