Re: sets definable by polynomials

( yet intresting i snip it )


Here are a few questions ...


not just a few , the most important id say.

im glad we agree on the basic questions.
( i waited for the idea to show up naturally )

i was wondering if a generalization would help or rather confuse ? ( meaning hypergeo in the integer input/output domain )

7 questions , sounds like gods pr-set questions :-)

anyways

(1) Do there exist two infinite pr-subsets of Z whose
union is not a
pr-set?

this question has been asked bye me before on another forum...

i believe i called it the AND problem.
( i like using 'and' 'or' 'not' in set theory , number theory , calculus , technology , logistics ...well math in general thus. it makes thinks as simple and logical as possible ( like einstein wanted ))

notice that if you change pr to non pr it is no longer neccesary true.

i believe this is the most important question , though probably equivalent to some of the other 6.

(2) If A is an infinite pr-subset of Z, and if B is a
finite subset of
Z, must (A union B) be a pr-set?

if all other 6 questions answers yes, i think it is unevitable.


(3) If A is an infinite pr-subset of Z, and if B is a
finite subset of
A, must (A\B) be a pr-set?

notice this is (1) and 'generalized'(7) together.
since \B is "deniel".


(4) If A,B are pr-subsets of Z such that A\B is
infinite, must A\B be
a pr-set?

again similar story.

if "deniel" exists and always is a polynomial,
than every polynomial is also already a "denied" state of the integers.

substracting a set B is then 'twice denied'

'twice denied ' is a union (1) pr-set if every seperate 'deniel' is polynomial.

and bye the assumed truth of (1) a union (if 2 pr-sets) is pr-set too.

and thus 'twice denied' ( = (4) ) is a pr-set too if (1) is true and "deny" (generalized (7)) is true.


(5) If A,B are pr-subsets of Z such that (A intersect
B) is infinite,
must (A intersect B) be a pr-set?

equivalent to (4) ( wich is equivalent to (1)+generalized(7) ) PLUS subsets (without intersect) of pr-sets / "deniels" are pr-sets too.


(6) If A is a pr-subset of Z, must {abs(a) | a in A}
be a pr-set?

i believe the answer is in the work of matheyasevich and i presume the primes are a counterexample ??

(primes are not pr-sets , but abs ( pr-sets) can be)
not sure though.

(7) [tommy1729's question] Is the set of non-square
positive integers
a pr-set?

this is then the "NOT" or "deniel" question in its general form.

i have had "and" ,"not" , but here is (logically)

"or" ->

is there always a pr-set for every A and B , where A and B are pr-sets too , which always gives an element of A or B but never elements that are in both (8), and never the same element twice (9)?

this is strongly related to the "deniel" of (5) with non-overlap restriction.

********

curiously is the example of the "deniel" of an exponential/polynomial equation :

"deny" 4^n (8m+7) = a^2 + b^2 + c^2

(which can also be written in reverse of course !!)

tommy1729

quasi
.