Re: -- more factoring

quasi wrote :

On Tue, 26 Feb 2008 07:29:35 EST, tommy1729
<tommy1729@xxxxxxxxx>
wrote:

quasi wrote:

Please state a complete and precisely worded
conjecture. Keep it
short and simple, but rigorously worded, and with
correct use of
quantifiers.

there exist distinct positive integers a and b
larger then 1 such
that gcd (a,b) = 1

and an integer polynomial f(x) exists that only has
its
primefactors 1 mod a

and an integer polynomial g(x) exists that only has
primefactors 1 mod b.

neither polynomials have primefactors 1 mod ab.

That's a lot better, but still vague in places.

sorry , i had to leave.



Focusing only on f, here's one way to word part of
your claim ...



Tommy's Conjecture:

There exists a nonconstant univariate integer
polynomials f, and
integers a,b > 1 with gcd(a,b) = 1, such that

For all POSITIVE integers n,

if p is prime and p|f(n)

then p = 1 (mod a) and p =/= 1 (mod b).



thanks for the restatement.

i added " POSITIVE " integers to be complete.

( without that "POSITIVE" my claim is probably false )



Remark: I don't believe the above conjecture. In
fact, I'll make two
counter-conjectures. The first one is a variant of
one I proposed a
few months ago. The other is the one I proposed
earlier in this
thread.

quasi's conjectures ...

Conjecture (1):

If f is a nonconstant univariate integer polynomial,
and a is an
integer, a > 1, then there exists an integer n and a
prime p such that
p|f(n) and p = 1 (mod a).

yes i know that.

but i cant give you credit for that , since

1) its not a counter-conjecture at all.

2) the idea is actually mine , about a year ago i posted this here. and i gave an example of primefactors 1 mod 10.
im not going to call that your conjecture thus.

its mine.

3) i agree on that.

4) weaker versions have already been proven , so i assume it is almost as good as a hypothesis or a theorem.

5) by the above arguments , the idea is far from new.

but at least we agree on it ;-)



Conjecture (2):

There do not exist nonconstant univariate integer
polynomials f,g such
that gcd(f(m),g(n)) = 1 for all m,n in Z.

thats the real counterconjecture.

in the OP i stated the exact opposite and the truth of " tommy's conjecture " on this page implies the opposite.



quasi

i hope all is clear now.

thanks for your patience.

regards
tommy1729
.

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