I was fascinated by Fermats conjecture that there are no integer numbers x,y,z and x·y·z >0 such than xn+yn=zn for n >2.
Richard Feynman tried to prove this conjecture: His idea was to use a probabilistic theory approach and show that the probability for a triple x,y,z to satisfy the equation is extremely small. Of course this is not a proof.
I started around 14 years old a similar approach trying to filter out combinations and hoping that at the end I can get all of them so that finally the conjecture is proved.
I observed that there is a property which I call the decimal sum L(x) or L-value of an integer x:
For example L(45634) = 4+5+6+3+4=22 and L(22) = 2+2 = 4. So finally L(45634) = 4. The L function is applied until a single digit comes out. I found that numbers with a L value = 1,4 or 7 form a subgroup under multiplication with the following Cayley table:
|
* |
1 |
4 |
7 |
|
1 |
1 |
4 |
7 |
|
4 |
4 |
7 |
1 |
|
7 |
7 |
1 |
4 |
Looking at the properties of this group I could proove that for integers a,b,c,n,m,k> 0
a2n + b2m = c2k can be only true if abc º0(mod3)
and also only true if abc º0(mod5)
The main group is formed under the multiplication * defined as L(a)*L(b) = L(a*b)
Its Cayley table is:
|
* |
1 |
2 |
4 |
5 |
7 |
8 |
|
1 |
1 |
2 |
4 |
5 |
7 |
8 |
|
2 |
2 |
4 |
8 |
1 |
5 |
7 |
|
4 |
4 |
8 |
7 |
2 |
1 |
5 |
|
5 |
5 |
1 |
2 |
7 |
8 |
4 |
|
7 |
7 |
5 |
1 |
8 |
4 |
2 |
|
8 |
8 |
7 |
5 |
4 |
2 |
1 |
Similar from this group it follows that if a º/º0(mod3) then L(a6n) = 1, for n>0
Other results are:
a4k+2 + b4m+2 = c4n+2 is not true if abcº/º0(mod5)
I have found other groups such as the numbers
(50k ± a)2 with a Î {4,6,14,16 ,24}
These numbers form a group under multipilcation. Also the number 4,6,14,16 and 24 form a cyclic abelian group under multiplication *. The unity element is 24.
|
* |
4 |
6 |
14 |
16 |
24 |
|
4 |
16 |
24 |
6 |
14 |
4 |
|
6 |
24 |
14 |
16 |
4 |
6 |
|
14 |
6 |
16 |
4 |
24 |
14 |
|
16 |
14 |
4 |
24 |
6 |
16 |
|
24 |
4 |
6 |
14 |
16 |
24 |
For example 4*14 = 6 since 4·14º6(mod50)
If we define the product a*b = c where abºc(mod50)
It follows that (50k ± a)10nº24(mod50)and L((50k ± a)10n) = 7.
Of course L(a) = b is equivalent to aºb(mod9) if a =/= 0
For all integer numbers a Î N can be written as a= 9n+i. Three sets L0, L1, and L2 can be then defined
L0 = {x | x º i(mod9) and iº 0(mod3) }
L1 = {x | x º i(mod9) and iº 1(mod3) }
L2 = {x | x º i(mod9) and iº 2(mod3) }
It follows that and L1 and L2 form a group and L1 is a subgroup. Therefore if x Î L1 or L2 then
x6n = 1 i.e
x6n º 1(mod3)
x6n º 1(mod9)
L0 forms a group under addition with the following Cayley table:
|
+ |
0 |
3 |
6 |
|
0 |
0 |
3 |
6 |
|
3 |
3 |
6 |
0 |
|
6 |
6 |
0 |
3 |
L(x) = j if x º j(mod9).
The addition is defined by the corresponding L value
Example:
For 12+5 = 17 we have L(12) + L(3) = 3+5 = L(17) = 8. Therefore we write 3+5=8
and multiplication
12·5 = 60
i.e.
L(12) L(5) = 3*5 = L(60) = 6. Therefore we write 3*5 = 6
It follows that
a6k+5 + b6l+5 =/= c6m+5 if a·b Î L2
Combining various such groups more complex properties can be found for multiplication and summation of powers.
A few years ago Wiles proved Fermat's theory using various group theoretical results and objects like modular forms and elliptic curves. It is strange how all these are connected with objects like the Monster sporadic group and finally with string theory. Finally if string theory is more than a very complex mathematical game then numbers and structures or symmetries seems to be the fundamental building blocks of nature. The whole thing is a number! Pythagoras would be very happy!
Finally matter dissapears and only structures and relations exist and one can only repeat the words of Alice in Wonderland:
"Well! I’ve often seen a cat without a grin," thought Alice;
"but a grin without a cat! It’s the most curious thing I ever saw in all my life!"