For given positive
Do there exist such that:
(is s a subset sum of the miís)?
Every prime that is 1mod4 can be
written as sum of two squares.
Those squares are found using LLL
For define the radical
(Thatís the product of distinct prime factors of a,b,c). suppose gcd(a,b,c)=1.
abc conjecture: For every x>1 there exists only finitely many a,b,c with gcd(a,b,c) = 1 and a + b = c such that
The search for examples uses LLL
B1,bn are reduced basis for a lattice L in b1*, bn* defined as before. Then:
(i.e. the 1st vector is ďreasonablyĒ short).
Reduced basis, what is it good for?
so each is a pos. real number
D changes only if some bi* is changed, which only occurs at case 1 of the algorith. The number is reduced by a factor of ¾ since is, while the other
diís are unchanged. Hence D reduced by factor of ¾ .
diís are bounded from below which bounds D from below.
So thereís an upper bound for # of times we pass through case 1.
In end of case 1, k = k-1
End of case 2, k = k+1
Start with k = 2, and
So # of times we pass through case 2
Is at most n-1 more than the # of times we pass through case 1,
Hence the algorith terminates.
Initialization step with rationales:
# of times pass through case 1:
# of times pass through case 2:
Case 1 requires operations
Case 2 we have values of p
Each requires operations
Hence we get a total of