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14
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Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)= p$, a prime. Let $g(x)$ and $h(x)$ have not all coefficients negative nor positive. Can $g(x)h(x)$ have only positive or only negative coefficients? Is there a way to generate such polynomials? What if $a < 1$? The motivation is to test if a given polynomial has only positive coefficients and assumes a prime value at a positive integer larger than $1$, then it is irreducible.
I got good answers for the above question. The answer includes the phrase "...the last few coefficients of $g(x)h(x)$ are large enough...". Is there a way to overcome this?
Adding one additional constraint:
The given polynomials $g(x)$ and $h(x)$ have constant coefficient $\pm 1$ with $g(x)h(x)$ having constant coefficient $1$, $g(a)h(a)$ a prime for some $a > 1$, say $a=2$.
Above also solved! My polynomials have one last constraint:
$x^{i}$ term coefficient of $g(x)h(x)$ cannot exceed say $2^{n−i+1}$ where $n$ is degree of $g(x)h(x)$ ($X^{0}$ TERM IS STILL $1$). The construction of $h_{1}(x)$ has large lower degree coefficients which means $h_{1}(x)$ has large higher degree coefficients. Is a factorization of the desired kind (unequal signs in coefficients) possible when $g(x)h(x)$ has this additional constraint? Again say $a=2$. The current construction by Ilya blows up the leading terms.
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13
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Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)= p$, a prime. Let $g(x)$ and $h(x)$ have not all coefficients negative nor positive. Can $g(x)h(x)$ have only positive or only negative coefficients? Is there a way to generate such polynomials? What if $a < 1$? The motivation is to test if a given polynomial has only positive coefficients and assumes a prime value at a positive integer larger than $1$, then it is irreducible.
I got good answers for the above question. The answer includes the phrase "...the last few coefficients of $g(x)h(x)$ are large enough...". Is there a way to overcome this?
Adding one additional constraint:
The given polynomials $g(x)$ and $h(x)$ have constant coefficient $\pm 1$ with $g(x)h(x)$ having constant coefficient $1$, $g(a)h(a)$ a prime for some $a > 1$, say $a=2$.
Above also solved! My polynomials have one last constraint:
$x^{i}$ term coefficient of $g(x)h(x)$ cannot exceed say $2^{n−i+1}$ where $n$ is degree of $g(x)h(x)$ ($X^{0}$ TERM IS STILL $1$). The construction of $h_{1}(x)$ has large lower degree coefficients which means $h_{1}(x)$ has large higher degree coefficients. Is a factorization of the desired kind (unequal signs in coefficients) possible when $g(x)h(x)$ has this additional constraint? Again say $a=2$. The current construction by Ilya blows up the leading terms.
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12
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On reducible polynomials with positive coefficientsand , $1$ as constant coefficient and certain bounds on coefficients
Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)= p$, a prime. Let $g(x)$ and $h(x)$ have not all coefficients negative nor positive. Can $g(x)h(x)$ have only positive or only negative coefficients? Is there a way to generate such polynomials? What if $a < 1$? The motivation is to test if a given polynomial has only positive coefficients and assumes a prime value at a positive integer larger than $1$, then it is irreducible.
I got good answers for the above question. The answer includes the phrase "...the last few coefficients of $g(x)h(x)$ are large enough...". Is there a way to overcome this?
Adding one additional constraint:
The given polynomials $g(x)$ and $h(x)$ have constant coefficient $\pm 1$ with $g(x)h(x)$ having constant coefficient $1$, $g(a)h(a)$ a prime for some $a > 1$, say $a=2$.
Above also solved! My polynomials have one last constraint:
$x^{i}$ term coefficient of $g(x)h(x)$ cannot exceed say $2^{n−i+1}$ where $n$ is degree of $g(x)h(x)$ ($X^{0}$ TERM IS STILL $1$). The construction of $h_{1}(x)$ has large lower degree coefficients which means $h_{1}(x)$ has large higher degree coefficients. Is a factorization of the desired kind (unequal signs in coefficients) possible when $g(x)h(x)$ has this additional constraint? Again say $a=2$.
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11
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10
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9
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Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)= p$, a prime. Let $g(x)$ and $h(x)$ have not all coefficients negative nor positive. Can $g(x)h(x)$ have only positive or only negative coefficients? Is there a way to generate such polynomials? What if $a < 1$? The motivation is to test if a given polynomial has only positive coefficients and assumes a prime value at a positive integer larger than $1$, then it is irreducible.
I got good answers for the above question. The answer includes the phrase "...the last few coefficients of $g(x)h(x)$ are large enough...". Is there a way to overcome this?
Adding one additional constraint:
The given polynomials $g(x)$ and $h(x)$ have constant coefficient $\pm 1$ with $g(x)h(x)$ having constant coefficient $1$.1$, $g(a)h(a)$ a prime for some $a > 1$, say $a=2$.
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8
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On reducible polynomials with positive coefficients and $1$ as constant coefficient
Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)= p$, a prime. Let $g(x)$ and $h(x)$ have not all coefficients negative nor positive. Can $g(x)h(x)$ have only positive or only negative coefficients? Is there a way to generate such polynomials? What if $a < 1$? The motivation is to test if a given polynomial has only positive coefficients and assumes a prime value at a positive integer larger than $1$, then it is irreducible.
I got good answers for the above question. The answer includes the phrase "...the last few coefficients of $g(x)h(x)$ are large enough...". Is there a way to overcome this?
Adding one additional constraint:
The given polynomials $g(x)$ and $h(x)$ have constant coefficient $\pm 1$ with $g(x)h(x)$ having constant coefficient $1$.
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7
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Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)= p$, a prime. Let $g(x)$ and $h(x)$ have not all coefficients negative nor positive. Can $g(x)h(x)$ have only positive or only negative coefficients? Is there a way to generate such polynomials? What if $a < 1$? The motivation is to test if a given polynomial has only positive coefficients and assumes a prime value at a positive integer larger than $1$, then it is irreducible.
I got good answers for the above question.
Adding an one additional constraint:
The given polynomials $g(x)$ have and $h(x)$ has have constant coefficient $\pm 1$ with $g(x)h(x)$ having constant coefficient $1$.
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6
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Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)= p$, a prime. Let $g(x)$ and $h(x)$ have not all coefficients negative nor positive. Can $g(x)h(x)$ have only positive or only negative coefficients? Is there a way to generate such polynomials? What if $a < 1$? The motivation is to test if a given polynomial has only positive coefficients and assumes a prime value at a positive integer larger than $1$, then it is irreducible.
I got good answers for the above question.
Adding an additional constraint: The given polynomials $g(x)$ have $h(x)$ has constant coefficient $\pm 1$ with $g(x)h(x)$ having constant coefficient $1$.
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5
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Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)=\pm h(a)= p$, a prime. Let $g(x)$ and $h(x)$ have not all coefficients negative nor positive. Can $g(x)h(x)$ have only positive or only negative coefficients? Is there a way to generate such polynomials? What if $a < 1$? The motivation is to test if a given polynomial has only positive coefficients and assumes a prime value at a positive integer larger than $1$, then it is irreducible.
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4
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3
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Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)=\pm p$, a prime. Let $g(x)$ and $h(x)$ have not all coefficients negative nor positive. Can $g(x)h(x)$ have only positive or only negative coefficients? Is there a way to generate such polynomials? What if $a < 1$? The motivation is to test if a given polynomial has only positive coefficients and assumes a prime value at a positive integer larger than $1$, then it is irreducible.
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2
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1
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On reducible polynomials
Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)=\pm p$, a prime. Let $g(x)$ and $h(x)$ have not all coefficients negative nor positive. Can $g(x)h(x)$ have only positive or only negative coefficients? Is there a way to generate such polynomials? What if $a < 1$?
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