We explore two specific connections between prime numbers and polynomials. Cohn's Criterion states that if $d_nd_{n-1}\\ldots d_0$ is the base $10$ representation of a prime, then the polynomial $\\sum_{j=0}^n d_jx^j$ is irreducible. Let $f(x)$ be a polynomial with non-negative integer coefficients. We define $c(10)$ be the largest integer such that if $f(10)$ is prime and all the coefficients of $f(x)$ are $\\le c(10)$, then $f(x)$ is irreducible. It is known that $$2.52~\\times~10^{30}\\le c(10)\\le 4.96~\\times~10^{31}.$$ We improve the lower bound above to $5.21~\\times~10^{30}$. Furthermore, we classify all reducible polynomials $f(x)$ with non-negative integer coefficients such that $f(10)$ is prime and all the coefficients of $f(x)$ are $\\le 4.96~\\times~10^{31}$. Let $S$ be a finite set of rational primes. For a non-zero integer $n$, define $\\left[ n\\right] _{S}=\\prod_{p\\in S}\\left\\vert n\\right\\vert _{p}^{-1}$% , where $\\left\\vert n\\right\\vert _{p}$ is the usual $p$-adic norm of $n$. In 1984, Stewart applied Baker's theorem to prove non-trivial, computationally effective upper bounds for $[n(n+1)...(n+k)]_S$ for any integer $k>0$. Effective upper bounds have also been given by Bennett, Filaseta, and Trifonov for $[n(n+1)]_S$ and $[n^2+7]_S$, where $S=\\{2,3\\}$ and $S=\\{2\\}$, respectively. We extend Stewart's theorem to prove effective upper bounds for $[f(n)]_S$ for an arbitrary $f(x)$ in $\\mathbb{Z}[x]$ having at least two distinct roots.
","abstract_html":"<p>We explore two specific connections between prime numbers and polynomials. Cohn's Criterion states that if dndn-1\\ldots d0 is the base $10$ representation of a prime, then the polynomial \\sumj=0n djxj is irreducible. Let $f(x)$ be a polynomial with non-negative integer coefficients. We define $c(10)$ be the largest integer such that if $f(10)$ is prime and all the coefficients of $f(x)$ are $\\le c(10)$, then $f(x)$ is irreducible. It is known that $2.52~\\times~1030\\le c(10)\\le 4.96~\\times~1031.$ We improve the lower bound above to $5.21~\\times~10^{30}$. Furthermore, we classify all reducible polynomials $f(x)$ with non-negative integer coefficients such that $f(10)$ is prime and all the coefficients of $f(x)$ are $\\le 4.96~\\times~10^{31}$. Let $S$ be a finite set of rational primes. For a non-zero integer $n$, define $\\left[ n\\right] _{S}=\\prod_{p\\in S}\\left\\vert n\\right\\vert _{p}^{-1}$% , where $\\left\\vert n\\right\\vert _{p}$ is the usual $p$-adic norm of $n$. In 1984, Stewart applied Baker's theorem to prove non-trivial, computationally effective upper bounds for $[n(n+1)...(n+k)]_S$ for any integer $k>0$. Effective upper bounds have also been given by Bennett, Filaseta, and Trifonov for $[n(n+1)]_S$ and $[n^2+7]_S$, where $S=\\{2,3\\}$ and $S=\\{2\\}$, respectively. We extend Stewart's theorem to prove effective upper bounds for $[f(n)]_S$ for an arbitrary $f(x)$ in $\\mathbb{Z}[x]$ having at least two distinct roots.</p>","abstract_has_math":true,"creators":["Gross, Samuel S."],"institution":null,"degree_name":"Ph.D.","degree_level":"Campus Access Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Michael Filaseta"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-01-01T08:00:00Z","date_published":"2012-01-01T08:00:00Z","updated_at":"2026-07-24T04:38:31Z","subjects":["Mathematics","Physical Sciences and Mathematics"],"languages":[],"rights":["© 2012, Samuel S. Gross"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarcommons.sc.edu/etd/1597","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Michael Filaseta"]},{"key":"dc:creator","label":"Author","values":["Gross, Samuel S."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Campus Access Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Physical Sciences and Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["© 2012, Samuel S. Gross"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarcommons.sc.edu/etd/1597"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We explore two specific connections between prime numbers and polynomials. Cohn's Criterion states that if $d_nd_{n-1}\\ldots d_0$ is the base $10$ representation of a prime, then the polynomial $\\sum_{j=0}^n d_jx^j$ is irreducible. Let $f(x)$ be a polynomial with non-negative integer coefficients. We define $c(10)$ be the largest integer such that if $f(10)$ is prime and all the coefficients of $f(x)$ are $\\le c(10)$, then $f(x)$ is irreducible. It is known that $$2.52~\\times~10^{30}\\le c(10)\\le 4.96~\\times~10^{31}.$$ We improve the lower bound above to $5.21~\\times~10^{30}$. Furthermore, we classify all reducible polynomials $f(x)$ with non-negative integer coefficients such that $f(10)$ is prime and all the coefficients of $f(x)$ are $\\le 4.96~\\times~10^{31}$. Let $S$ be a finite set of rational primes. For a non-zero integer $n$, define $\\left[ n\\right] _{S}=\\prod_{p\\in S}\\left\\vert n\\right\\vert _{p}^{-1}$% , where $\\left\\vert n\\right\\vert _{p}$ is the usual $p$-adic norm of $n$. In 1984, Stewart applied Baker's theorem to prove non-trivial, computationally effective upper bounds for $[n(n+1)...(n+k)]_S$ for any integer $k>0$. Effective upper bounds have also been given by Bennett, Filaseta, and Trifonov for $[n(n+1)]_S$ and $[n^2+7]_S$, where $S=\\{2,3\\}$ and $S=\\{2\\}$, respectively. We extend Stewart's theorem to prove effective upper bounds for $[f(n)]_S$ for an arbitrary $f(x)$ in $\\mathbb{Z}[x]$ having at least two distinct roots.
"]},{"key":"dc:title","label":"Title","values":["Irreducibility Criteria For Polynomials With Non-Negative Integer Coefficients, and the Prime Factorization of F(N) For F(X) In Z[X]"]}]}],"canonical_facts":{"dc:contributor":["Michael Filaseta"],"dc:creator":["Gross, Samuel S."],"dc:description.abstract":["We explore two specific connections between prime numbers and polynomials. Cohn's Criterion states that if $d_nd_{n-1}\\ldots d_0$ is the base $10$ representation of a prime, then the polynomial $\\sum_{j=0}^n d_jx^j$ is irreducible. Let $f(x)$ be a polynomial with non-negative integer coefficients. We define $c(10)$ be the largest integer such that if $f(10)$ is prime and all the coefficients of $f(x)$ are $\\le c(10)$, then $f(x)$ is irreducible. It is known that $$2.52~\\times~10^{30}\\le c(10)\\le 4.96~\\times~10^{31}.$$ We improve the lower bound above to $5.21~\\times~10^{30}$. Furthermore, we classify all reducible polynomials $f(x)$ with non-negative integer coefficients such that $f(10)$ is prime and all the coefficients of $f(x)$ are $\\le 4.96~\\times~10^{31}$. Let $S$ be a finite set of rational primes. For a non-zero integer $n$, define $\\left[ n\\right] _{S}=\\prod_{p\\in S}\\left\\vert n\\right\\vert _{p}^{-1}$% , where $\\left\\vert n\\right\\vert _{p}$ is the usual $p$-adic norm of $n$. In 1984, Stewart applied Baker's theorem to prove non-trivial, computationally effective upper bounds for $[n(n+1)...(n+k)]_S$ for any integer $k>0$. Effective upper bounds have also been given by Bennett, Filaseta, and Trifonov for $[n(n+1)]_S$ and $[n^2+7]_S$, where $S=\\{2,3\\}$ and $S=\\{2\\}$, respectively. We extend Stewart's theorem to prove effective upper bounds for $[f(n)]_S$ for an arbitrary $f(x)$ in $\\mathbb{Z}[x]$ having at least two distinct roots.
"],"dc:identifier":["https://scholarcommons.sc.edu/etd/1597"],"dc:rights":["© 2012, Samuel S. Gross"],"dc:subject":["Mathematics","Physical Sciences and Mathematics"],"dc:title":["Irreducibility Criteria For Polynomials With Non-Negative Integer Coefficients, and the Prime Factorization of F(N) For F(X) In Z[X]"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Campus Access Dissertation"],"thesis:degree_name":["Ph.D."]},"updated_at":"2026-07-24T04:38:31Z"}