Re: Conjecture (FLT related): another karzeddin-like conjecture

On Sat, 22 Apr 2006 17:58:30 EDT, bassam king karzeddin
<bassam@xxxxxxxxxx> wrote:

quasi wrote:

The following conjecture was inspired by Karzeddin's
posts ...

Conjecture:

If a,b are positive integers greater than 2, and x,y
are coprime positive integers greater than 1, then
the prime factorization of x^a+y^b includes at least
one prime factor with exponent at most 2.

The same is true for x^a-y^b with the two exceptions
x=2,a=3,y=3,b=2 and x=3,b=2,y=2,a=3.

Remarks:

The truth of the above conjecture would yield an
instant alternative proof of FLT.

In fact, the above conjecture also implies the truth
of The Beal Conjecture.

I'm happy that Quasi's conjectures are still challenging,
not only because they support mine, but also because
they are very enlighting

I will be more happy if any body can refute them or prove
them, because then, we can learn something new

These conjectures are easy to make, hard to prove. In a sense math is
not yet ready for these types of questions.

What these questions have in common is that they can all be viewed as
asking whether or not two low density sets have a nonempty
intersection.

Thus, for my conjecture above:

Let A = {n in N | n=x^a +y^b, where a,b are positive integers and x,y
are coprime positive integers greater than 1}

Let B = {n in N | n>1 and, for every prime factor p of n, p^3 also
divides n}.

The first part of my conjecture is equivalent to the claim that A and
B have no common elements.

The fact is, both of the sets A, B are sparse and get sparser
(although I'm not sure about that). Thus, density based probability
arguments suggest that, all other things being equal, the sets A,B
have a positive probability of having an empty intersection. The
density argument doesn't yield a proof that the intersection is empty,
but it does help explain why counterexamples can't be easily produced.

I discuss these ideas further in the thread:

"probabilities based on asymptotic densities"

quasi
.

Relevant Pages

  • Re: Computability
    ... the readers. ... Given that x,y,z are now assumed as positive integers, ... Suggestive of the truth of the conjecture? ... proposition true does, however, prove FLT by a new route, one not ...
    (sci.math)
  • Re: Computability
    ... the readers. ... Given that x,y,z are now assumed as positive integers, ... The converse to what statement? ... Suggestive of the truth of the conjecture? ...
    (sci.math)
  • Things get nicer at some huge number?
    ... Mathematicians have long noticed that in many fields, ... exceptions for small integers. ... THE BIG NUMBER CONJECTURE CONJECTURE (sic) ... ranges over the positive integers, then there is a number B_P ...
    (sci.math.research)
  • Re: Difference of like powers of rational numbers
    ... for all sufficiently large positive integers n ... Conjecture 1c: ... If x,y are distinct nonzero complex numbers such that ... If S is infinite, then S contains an infinite arithmetic progression. ...
    (sci.math)
  • Re: seeking simple integer number games
    ... If the original conjecture pertained to positive integers ... Because cycles are determined solely by ... of whether 'a' and 'g' are in the positive or negative domain. ... > superset of the positive integers, then you have proven it, but showing ...
    (sci.math)
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