The following paragraphs contain a short outlook on the Last Fermat theorem with regards to G-systems.
Let c(s) be a number of instances in segment s. In case a^p +b^p =c^p (i.e. the Last Fermat theorem) it should hold:
c-1
(1) ∑(c(s)) = a^p
s=b
For example in G(p) the values c(s) are:
p=2: 1, 3, 5, 7, 9, 11 13, 15, 17, 19, 21, 23, 25, 27 .. p=3: 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, ... p=5: 1, 31, 211, 781, 2101, 4651, ... p=7: 1, 127, 2059, 14197, ...
Only in case p=2 such sums are known:
7+ 9 = 4^2; 17+19 = 6^2; 11+13+15+17+19+21+23+25 = 12^2.
In case p=3, i.e. G(3) the values c(s) are:
| 0 1 2 3 4 ... i --+---------------------------- 0 | 0 1 | 1, 7, 19, 37, 61, 91 2 | 8, 26, 56, 98,152, 3 | 27, 63,117,189, 4 | 64,124,208, 5 | 125, 6 | j
In the table it holds:
(2a) R[i,j]=R[i,1]+R[i+1,j-1] E.g. R[3,3]=R[3,1]+R[4,2]=19+98=117
Written in an other way:
(2b) R[i,j]=R[i-1,j+1]-R[i-1,1] E.g. R[4,2]=R[3,3]-R[3,1]=117-19=98
The equation (1) must hold also for every module m:
c(s)
mod2 mod3 mod4 mod5 mod6 mod7 mod8 mod9 mod10 mod11 mod12
-----------------------------------------------------------------
1 1 1 1 1 1 1 1 1 1 1 1
7 1 1 -1 2 1 0 -1 -2 -3 -4 -5
19 1 1 -1 -1 1 -2 3 1 -1 -3 -5
37 1 1 1 2 1 2 -3 1 -3 4 1
61 1 1 1 1 1 -2 -3 -2 1 -5 1
91 1 1 -1 1 1 0 3 1 1 3 -5
127 1 1 -1 2 1 1 -1 1 -3 -5 -5
169 1 1 1 -1 1 1 1 -2 -1 4 1
217 1 1 1 2 1 0 1 1 -3 -3 1
271 1 1 -1 1 1 -2 -1 1 1 -4 -5
331 1 1 -1 1 1 2 3 -2 1 1 -5
397 1 1 1 2 1 -2 -3 1 -3 1 1
469 1 1 1 -1 1 0 -3 1 -1 -4 1
547 1 1 -1 2 1 1 3 -2 -3 -3 -5
a^p
mod2 mod3 mod4 mod5 mod6 mod7 mod8 mod9 mod10 mod11 mod12
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1 1 1 1 1 1 1 1 1 1 1 1
8 0 -1 0 -2 2 1 0 -1 -2 -3 -4
27 1 0 -1 2 3 -1 3 0 -3 5 3
64 0 1 0 -1 -2 1 0 1 4 -2 4
125 1 -1 1 0 -1 -1 -3 -1 5 4 5
216 0 0 0 1 0 -1 0 0 -4 -4 0
343 1 1 -1 -2 1 0 -1 1 3 2 -5
512 0 -1 0 2 2 1 0 -1 2 -5 -4
729 1 0 1 -1 3 1 1 0 -1 3 -3
1000 0 1 0 0 -2 -1 0 1 0 -1 4
1331 1 -1 -1 1 -1 1 3 -1 1 0 -1
1728 0 0 0 -2 0 -1 0 0 -2 1 0
2197 1 1 1 2 1 -1 -3 1 -3 -3 1
2744 0 -1 0 -1 0 0 0 -1 4 5 -4
Therefore, sum of numbers in a block of some adjacent rows from the first table should be equal to the numbers of one row from the second table (for each column).
From the expression (b+p)^p-b^p = 0 (mod p) and from the structure of diferential progressions of self-classes follows:
s[(b+p)^p-(b+p)]/p - (b^p-b)/p = -1 (mod p )
(b+p)^p - b^p = 0 ( mod p^2 ).
E.g. (2+3)^3 - 2^3 = 117 = 0 mod 3^2; or (2+5)^5 - 2^5 = 16775 = 0 mod 5^2.
For each p ε P the numbers c(s) satisfy the equation:
a(0) + a(p) = constant ( mod p ).
E.g. p=3: 1 + 37 = 7 + 61 = 19 + 91 = 37 + 127 = 2 mod 3.
Let us search all sequences r= {r(i)} having members r(a),r(b),r(c), that
it holds: r(a)+r(b)=r(c).
Fermat theorem only reduces domain of sequences to r(n)= n^k (k is a constant).