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## Polynomial and primes

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• It is an old problem in this forum, how much primes a polynomial does represent. Admissible irreducible polynomials with integral coefficients and positive
Message 1 of 2 , Dec 5, 2006
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It is an old problem in this forum, how much primes

a polynomial does represent.

Admissible irreducible polynomials with integral

coefficients and positive leading coefficients,

if it represents a prime, then it represents infinitely many primes.

This is a party of Schinzel's hypothesis. I am liberty

to attack it in the paper “There are infinitely many primes of the form a^{2}+1”.

Welcome point out any error.

Original Schinzel's hypothesis: Admissible irreducible polynomials with

integral coefficients and positive leading coefficients represent infinitely many primes.

It seem this involved the Ross-Littwood paradox, which is a contradiction between

numerical limit (not equal 0) and set theoretic limit(empty), like Phil has pointed out.

I think that it is impossible to prove it.

Example: we have not proved that there is number a > 41 such that

x^2 �Cx +a represents primes for x=0,1,2,…,a. If we find such number a or prove there

is such number a, then there are infinitely many number x such that x^2 �Cx +a represents

primes for x=0,1,2,…,a.

I formalized Ross-Littwood paradox in second order formal system P(N), can use

prime patterns to abstain the paradox.

If delete condition “irreducible”, then we say that a composite polynomial,

except the finite primes of the form 1*f_2(x), has not prime patterns of the

form f_1(x)*f_2(x) for f_1(x)>1 and f_2(x) >1.

We have a conclusion:

Admissible polynomials with integral coefficients and positive leading coefficients,

if it has a prime pattern, then it represents infinitely many primes.

Welcome anyone continues discussion of this problem.

I admit my mistake and revise it. Thank Kermit.

Liu Fengsui.

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Mp3疯狂搜-新歌热歌高速下

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• ... If I understand correctly, then the proof mentioned at http://mathworld.wolfram.com/LuckyNumberofEuler.html shows that there is no number 41 that works
Message 2 of 2 , Dec 5, 2006
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On 12/5/06, 逢绥 刘 <liu_f_s@...> wrote:
> Example: we have not proved that there is number a > 41 such that
>
> x^2 �Cx +a represents primes for x=0,1,2,…,a. If we find such number a or prove there
>
> is such number a, then there are infinitely many number x such that x^2 �Cx +a represents
>
> primes for x=0,1,2,…,a.

If I understand correctly, then the proof mentioned at
http://mathworld.wolfram.com/LuckyNumberofEuler.html
shows that there is no number > 41 that works here (though of course
you meant not to include a in the list, since a^2 - a + a is surely
not prime).

--Joshua Zucker
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