A multigrade is an equality relationship between two (or more) groups of numbers where the sum of their powers is equal for two or more distinct, positive powers (I first became aware of them through Albert Beiler's "Recreations In The Theory Of Numbers").
Multigrades can be asymmetric or symmetric with respect to their terms. An example of an asymmetric multigrade is 3^n + 3^n = 1^n + 1^n + 4^n which is true for n = 1,2. From all the examples of I've examined, asymmetric multigrades differ by at most one term so for the present example, you have two terms on one side of the equation and three terms on the other side.
However asymmetry is only is a mirage because all asymmetric multigrades can be converted into symmetric ones through the addition of 0. So by adding 0, the above equation is changed into 0^n + 3^n + 3^n = 1^n +1^n + 4^n, n = 1,2.
GDM means Gauss's Dot Multiplier meaning you're dot multiplying by consecutive numbers, starting from 1 on the left side of the equation and dot multiplying again, starting from 1, on the right side of the equation. So for the top equation, multiplying by GDM yields 1 x 3^n + 2 x 3^n for the left side of the equation and 1 x 1^n + 2 x 1^n + 3 x 4^n on the right side of the equation. Note that there is no equality for any value of n.
The importance of the 0 term will now be shown. Plug in 0 to get the second equation above and apply GDM to the second equation to get:
1 x 0^n + 2 x 3^n + 3 x 3^n = 1 x 1^n + 2 x 1^n + 3 x 4^n which is true for n = 1. With trigrades, I've found n = 1,2 can work when I applied the GDM (but there is no guarantee that applying GDM will work with all multigrades).
In July I started to go to higher multigrades (tetragrades, pentagrades, etc.), but n never exceeded 2 after I applied the GDM. This made me think of FLT which I believe Media_Man and Tonio is beginning to understand.
There is much more to multigrades. You can have multigrades with more than one equal sign. You can have alternating multigrades which are true for either even or odd powers of n. I have an example of a multigrade which is true for n = 2,3,4, but not 1. I suspect you can have multigrades that skip powers in other ways (e.g. n = 1,4,7,10...).
I hope in general that all visitors can see what I'm getting at. I'll be dropping from by to by. Thank you you Media_Man for your computer help
as two hours a day at the public library doesn't cut it and thank you Tonio for looking further. I'll do one more thing on this thread.
Take the MS:
16 2 3 13
5 11 10 8
9 7 6 12
4 14 15 1
The magic sum is 34. Take the diagonals 3,9,11 and 6,8,14. When you double the middle numbers, you get 3,11,11,9 and 6,6,8,14 which is a bigrade. Have fun