Re: a simple concept FACTORING question 1001
- From: quasi <quasi@xxxxxxxx>
- Date: Wed, 17 Oct 2007 23:24:05 -0400
On Wed, 17 Oct 2007 23:20:49 -0400, quasi <quasi@xxxxxxxx> wrote:
On Thu, 18 Oct 2007 02:35:52 GMT, Gerry Myerson
<gerry@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
In article <q0edh3d1jtc45d3q0h388etsri6bd9sp8g@xxxxxxx>,
quasi <quasi@xxxxxxxx> wrote:
On Tue, 16 Oct 2007 18:24:44 EDT, tommy1729 <tommy1729@xxxxxxxxx>
wrote:
all functions used are polynomials with rational
coefficients and positive integer valued for all
positive integer imput. (N0 -> N0)
all variables are positive integers.
g(x) is not a factor of f(x)
(factor in the sense of polynomial factors like
(x+1)(x-1))
but for every x > A, there exists a g(y) that is a
factor of f(x).
(weak version of the idea)
in fact all factors of f(x) are of the type g(y).
(stronger version of the idea)
an example with g(y) = 10y + 1 has already been given
by me before.
see "factoring tricks" where a degree 4 polynomial
has been given that has as primefactors exclusively
primes 1 mod 10.
thereby satisfying the stronger version (and the
weaker too of course)
now the question becomes
(weaker)
1)
a) does there exist a g(y) = y^2 + 1 and an f(x) such
that the weaker idea holds ?
b) does such a g(y) of irred degree 2 exist at all ?
c) does such an irred g(y) > degree 1 exist at all ?
(stronger)
2)
assuming it is already proved that g(y) and f(x) both
generate an infinite amount of primes;
a) does there exist a g(y) = y^2 + 1 and an f(x) such
that the stronger idea holds ?
b) does such a g(y) of irred degree 2 exist at all ?
c) does such an irred g(y) > degree 1 exist at all ?
regards
tommy1729
sorry if its a dumb question
Those are definitely not dumb questions.
For question 1(a), I'll conjecture a "yes" answer, but I'm not sure.
Of course, a "yes" answer to question 1(a) would also imply a "yes"
answer to all parts of question (1).
For question 2(c), I'll conjecture a "no" answer, and I'm almost sure
of it. Moreover, I think I can prove it, but I probably won't have
time to play with it until the weekend. Of course, a "no" answer to
question 2(c) would also imply a "no" answer to all parts of question
(2).
Maybe I'm misunderstanding the question,
but if f(x) = x^2 + x + 2 and g(y) = y^2 + 1
then for every x f(x) has g(1) as a factor.
Oops -- in my quick reading of the problem, I missed the order of the
quantifiers.
As stated, of course you're right, and thus question (1) is trivial.
How about this alternate version of tommy's question ...
Question 1 (a) [alternate version]:
Does there exist an irreducible integer polynomial f(x) of degree
greater than 2 such that, for all integers b, there exists an integer
b such that f(a) is a nonzero multiple of b^2 + 1?
correction:
such that, for all integers b, there exists an integer
a such that f(a) is a nonzero multiple of b^2 + 1?
Also, here's my slightly modified version of tommy's question 2(a):
Question 2 (a) [modified version]:
Does there exist a nonconstant integer polynomial f(x) such that, for
all integers a with f(a) nonzero, all prime factors of f(a) have the
form b^2 + 1 for some integer b?
quasi
.
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