On an Irreducibility Theorem of A. Cohn

Brillhart, John; Filaseta, Michael; Odlyzko, Andrew

doi:10.4153/cjm-1981-080-0

Cited by 34 publications

(31 citation statements)

References 2 publications

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“…Indeed, if / = gh is a non-trivial factorization of / in Z[X] then g(rn)h(rri) = f(m) = n is a non-trivial Splitting of n in Z. This result follows from the proofs in [4], where we need only the easier case m > 3. We note that / can be factored in time (logn)O(1\ by means of the algorithm of [21].…”

Section: Finding a Polynomialmentioning

confidence: 50%

Factoring integers with the number field sieve

Buhler

Lenstra

Pomerance

1993

Lecture Notes in Mathematics

146

171

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Abstract. In 1990, the ninth Fermat number was factored into primes by means of a new algorithm, the "number field sieve", which was invented by John Pollard. The present paper is devoted to the description and analysis of a more general version of the number field sieve. It should be possible to use this algorithm to factor arbitrary integers into prime factors, not just integers of a special form like the ninth Fermat number. Under reasonable heuristic assumptions, the analysis predicts that the time needed by the general number field sieve to factor n is exp((c+o(l))(logn)1/3(loglogn)2/3) (for n-*oo), where c=(64/9)l/3 = 1.9223. This is asymptotically faster than all other known factoring algorithms, such äs the quadratic sieve and the elliptic curve method. There does not yet exist an Implementation of the number field sieve for general integers, so that a practical comparison cannot yet be made.

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Section: Finding a Polynomialmentioning

confidence: 50%

Factoring integers with the number field sieve

Buhler

Lenstra

Pomerance

1993

Lecture Notes in Mathematics

146

171

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“…There are several results in the literature (see [7], [8], [9], and the references in those papers) which say that under appropriate conditions, if f(k) is prime, then/(x) is irreducible. (In [9], f(k) has to be prime for several values of k, actually.)…”

mentioning

confidence: 99%

Irreducibility Testing and Factorization of Polynomials

Adleman¹,

Odlyzko²

1983

Mathematics of Computation

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Abstract. It is shown that under certain hypotheses, irreducibility testing and factorization of polynomials with integer coefficients are polynomial time reducible to primality testing and factorization of integers, respectively. Combined with recently discovered fast primality tests, this yields an almost polynomial time irreducibility algorithm. The assertions of irreducibility produced by this algorithm are always certain and yield short proofs of irreducibility.

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“…Indeed, the derivative of this function is 3x 2 − 2x = x(3x − 2), revealing that The foregoing proof of Theorem 2 is really a motivated account of the proof in [1], where the authors adapted the method indicated in [5, p. 133] to deal with the general base. Our approach has been more naive and slightly different.…”

Section: Lemma 3 Suppose That α Is a Complex Root Of A Polynomialmentioning

confidence: 99%

“…This problem was subsequently generalized to any base b by Brillhart, Filaseta, and Odlyzko [1]. We will give a proof of this fact that is conceptually simpler than the one in [1], as well as study the analogue of this question for function fields over finite fields. More precisely, let F q denote the finite field of q elements, where q is a prime power.…”

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confidence: 99%

Prime Numbers and Irreducible Polynomials

Murty¹

2002

The American Mathematical Monthly

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The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry. There are certain conjectures indicating that the connection goes well beyond analogy. For example, there is a famous conjecture of Buniakowski formulated in 1854 (see Lang [3, p. 323]), independently reformulated by Schinzel, to the effect that any irreducible polynomial f (x) in Z [x] such that the set of values f (Z + ) has no common divisor larger than 1 represents prime numbers infinitely often. In this instance, the theme is to produce prime numbers from irreducible polynomials. This conjecture is still one of the major unsolved problems in number theory when the degree of f is greater than one. When f is linear, the conjecture is true, of course, and follows from Dirichlet's theorem on primes in arithmetic progressions.It is not difficult to see that the converse of the Buniakowski conjecture is true; namely, if a polynomial represents prime numbers infinitely often, then it is an irreducible polynomial. To see this, let us try to factorof positive degree. The fact that f (x) takes prime values infinitely often implies that either g(x) or h(x) takes the value ±1 infinitely often. This is a contradiction, for a polynomial of positive degree can take a fixed value only finitely often.There is a stronger converse to Buniakowski's conjecture that is easily derived (see . We will give a proof of this fact that is conceptually simpler than the one in [1], as well as study the analogue of this question for function fields over finite fields. More precisely, let F q denote the finite field of q elements, where q is a prime power. Fix a polynomialThe analog of Cohn's theorem to be proved in what follows is thatis irreducible in F q [t, x]. The proof in the function field case is much simpler and is motivated by the following elementary result, which can be viewed as somewhat of a strong converse to the conjecture of Buniakowski.

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On an Irreducibility Theorem of A. Cohn

Cited by 34 publications

References 2 publications

Factoring integers with the number field sieve

Factoring integers with the number field sieve

Irreducibility Testing and Factorization of Polynomials

Prime Numbers and Irreducible Polynomials

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