The Schinzel hypothesis claims (but it seems hopeless to prove) that any irreducible Q[x] polynomial without a constant factor assumes infinitely many prime values at integer places. On the other hand, it is easy to see that a reducible Q[x] polynomial can have only finitely many such places. In this paper we prove that a reducible Z[x] polynomial of degree n (where n is not 4 or 5) can have at most n+2 such places, and there exist examples with n+1 places. If the Schinzel hypothesis and the k-prime-tuple conjecture are true, then there are also polynomials with n+2 such places. (For n=4 or 5 the maximum possible value is 8.) If a Z[x] polynomial is the product of two non-constant integer-valued Q[x] polynomials, then there are at most 1.87234...n+o(n) such places. Even in this case, the number of integer places with positive prime values is at most n. More generally, if f=gh \in R[x] is a product of two non-constant real polynomials, then the number of real places x such that |g(x)|=1 or |h(x)=1|, while f(x)>1, is at most n. For a natural complex version of this last statement we give a counterexample. Furthermore, we briefly consider the generalized problem of assuming values in a fixed finite set of integers.
We adopt A. J. Irving's sieve method to study the almost-prime values produced by products of irreducible polynomials evaluated at prime arguments. This generalizes the previous results of Irving and Kao, who separately examined the almost-prime values of a single irreducible polynomial evaluated at prime arguments.
Let $\ell \geq 5$ be a prime and let $N$ be a square-free integer prime to $\ell$. For each prime $p$ dividing $N$, let $a_p$ be either $1$ or $-1$. We give sufficient criteria for the existence of a newform $f$ of weight 2 for $Γ_0(N)$ such that the mod $\ell$ Galois representation attached to $f$ is reducible and $U_p f = a_p f$ for primes $p$ dividing $N$. The main techniques used are level raising methods based on an exact sequence due to Ribet.
Let f in Z[x,y] be a reducible homoegeneous polynomial of degree 3. We show that f(x,y) has an even number of prime factors as often as an odd number of prime factors.
We give a method of constructing polynomials of arbitrarily large degree irreducible over a global field F but reducible modulo every prime of F. The method consists of finding quadratic f in F[x] whose iterates have the desired property, and it depends on new criteria ensuring all iterates of f are irreducible. In particular when F is a number field in which the ideal (2) is not a square, we construct infinitely many families of quadratic f such that every iterate f^n is irreducible over F, but f^n is reducible modulo all primes of F for n at least 2. We also give an example for each n of a q…
Let $K$ be a global field and $n > 1$ an integer. We show $n$ is composite if and only if there is an irreducible polynomial $f(x) \in K[x]$ of degree $n$ which is reducible $q$-adically for all the primes $q$ of $K$.
We calculate admissible values of r such that a square-free polynomial with integer coefficients, no fixed prime divisor and irreducible factors of degree at most 3 takes infinitely many values that are a product of at most r distinct primes.
For a given set of integers $\mathcal{S}$, let $\mathcal{R}_n^*(\mathcal{S})$ denote the set of reducible polynomials $f(X)=a_nX^n+a_{n-1}X^{n-1}+\cdots+a_1X+a_0$ over $\mathbb{Z}[X]$ with $a_i\in\mathcal{S}$ and $a_0a_n\ne 0$. In this note, we shall give an explicit bound of $|\mathcal{R}_n^*(\mathcal{S})|$. We also present an application of this bound to reducible bivariate polynomials over $\mathbb{Z}[X,Y]$.
Hooley proved that if $f\in \Bbb Z [X]$ is irreducible of degree $\ge 2$, then the fractions $\{ r/n\}$, $0<r<n$ with $f(r)\equiv 0\pmod n$, are uniformly distributed in $(0,1)$. In this paper we study such problems for reducible polynomials of degree $2$ and $3$ and for finite products of linear factors. In particular, we establish asymptotic formulas for exponential sums over these normalized roots.
We give an asymptotic formula for the divisor sum $\sum_{c<n\leq N}τ\left((n-b)(n-c)\right)$ for integers $b<c$ of the same parity. Interestingly, the coefficient of the main term does not depend on the discriminant as long as it is a full square. We also provide effective upper bounds of the average divisor sum for some of the reducible quadratic polynomials considered before, with the same main term as in the asymptotic formula.
We study root separation of reducible monic integer polynomials of odd degree. Let h(P) be the naive height, sep(P) the minimal distance between two distinct roots of an integer polynomial P(x) and sep(P)=h(P)^{-e(P)}. Let e_r*(d)=limsup_{deg(P)=d, h(P)-> +infty} e(P), where the limsup is taken over the reducible monic integer polynomials P(x) of degree d. We prove that e_r*(d) <= d-2. We also obtain a lower bound for e_r*(d) for d odd, which improves previously known lower bounds for e_r*(d) when d = 5, 7, 9.
Let $E$ be an elliptic curve over an imaginary quadratic field $K$, and $p$ be an odd prime such that the residual representation $E[p]$ is reducible. The $μ$-invariant of the fine Selmer group of $E$ over the anticyclotomic $\mathbb{Z}_p$-extension of $K$ is studied. We do not impose the Heegner hypothesis on $E$, thus allowing certain primes of bad reduction to decompose infinitely in the anticyclotomic $\mathbb{Z}_p$-extension. It is shown that the fine $μ$-invariant vanishes if certain explicit conditions are satisfied. Further, a partial converse is proven.
Let $p$ be an odd prime and $q$ a power of $p$. We examine the deformation theory of reducible and indecomposable Galois representations $\barρ:G_{\mathbb{Q}}\rightarrow \text{GSp}_{2n}(\mathbb{F}_q)$ that are unramified outside a finite set of primes $S$ and whose image lies in a Borel subgroup. We show that under some additional hypotheses, such representations have geometric lifts to the Witt vectors $\text{W}(\mathbb{F}_q)$. The main theorem extends that of Hamblen and Ramakrishna in which the $n=1$ case was treated.
Given a field $K$ and $n > 1$, we say that a polynomial $f \in K[x]$ has newly reducible $n$th iterate over $K$ if $f^{n-1}$ is irreducible over $K$, but $f^n$ is not (here $f^i$ denotes the $i$th iterate of $f$). We pose the problem of characterizing, for given $d,n > 1$, fields $K$ such that there exists $f \in K[x]$ of degree $d$ with newly reducible $n$th iterate, and the similar problem for fields admitting infinitely many such $f$. We give results in the cases $(d,n) \in \{(2,2), (2,3), (3,2), (4,2)\}$ as well as for $(d,2)$ when $d \equiv 2 \bmod{4}$. In particular, we show that for all…
We examine the question of when a quadratic polynomial f(x) defined over a number field K can have a newly reducible nth iterate, that is, f^n(x) irreducible over K but f^{n+1}(x) reducible over K, where f^n denotes the nth iterate of f. For each choice of critical point γof f(x), we consider the family g_{γ,m}(x)= (x - γ)^2 + m + γ, m \in K. For fixed n \geq 3 and nearly all values of γ, we show that there are only finitely many m such that g_{γ,m} has a newly reducible nth iterate. For n = 2 we show a similar result for a much more restricted set of γ. These results complement those obtained…
In a recent paper, Lapkova uses a Tauberian theorem to derive the asymptotic formula for the divisor sum $\sum_{n \leq x} d( n (n+v))$ where $v$ is a fixed integer and $d(n)$ denotes the number of divisors of $n$. We reprove her result by following a suggestion of Hooley, namely investigating the relationship between this sum and the well-known sum $\sum_{n \leq x} d( n ) d (n+v)$. As such, we are able to furnish additional terms in the asymptotic formula.
We prove that every odd semisimple reducible (2-dimensional) mod l Galois representation arises from a cuspidal eigenform. In addition, we investigate the possible different types (level, weight, character) of such a modular form. When the representation is the direct sum of the trivial character and a power of the mod l cyclotomic character, we are able to characterize the primes that can arise as levels of the associated newforms. As an application, we determine a new explicit lower bound for the highest degree among the fields of coefficients of newforms of trivial Nebentypus and prime leve…
Let Y_n(t) denote the set of all polynomials over the ring Z which are reducible over the field Q and of degree n>1 and of height not greater than t. We show that the true order of magnitude of |Y_n(t)| equals t^2 log t in the special case n=2 and it equals t^n for each n>2. We also determine the true order of magnitude of the size of certain interesting subsets of Y_n(t).
We give upper bounds on the numbers of various classes of polynomials reducible over the integers and over integers modulo a prime and on the number of matrices in SL(n), GL(n) and Sp(2n) with reducible characteristic polynomials, and on polynomials with non-generic Galois groups. We use our result to show that a random (in the appropriate sense) element of the mapping class group of a closed surface is pseudo-Anosov, and that a random automorphism of a free group is strongly irreducible (aka irreducible with irreducible powers). We also give a necessary condition for all powers of an algebrai…
A Diophantine $m$-tuple is a set $A$ of $m$ positive integers such that $ab+1$ is a perfect square for every pair $a,b$ of distinct elements of $A$. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by $x$. In doing so, we extend two existing tools in ways that might be of independent interest. The Erd\H os-Turán inequality bounds the discrepancy between the number of elements of a sequence that lie in a particular interval modulo 1 and the expected number; we establish a version of this inequality where the interval is allowed to vary. We also…
Let $l$ and $m$ be two integers with $l>m\ge 0$, and let $f(x)$ be the product of two linear polynomials with integer coefficients. In this paper, we show that $\log {\rm lcm}_{mn<i\le ln}\{f(i)\}=An+o(n)$, where $A$ is a constant depending only on $l$, $m$ and $f$.
For a degree $n$ polynomial $f$ over the rationals, the elements in the fiber $f^{-1}(a)$ are of degree $n$ over $\mathbb Q$ for most rational values $a$ by Hilbert's irreducibility theorem. Determining the set of exceptional $a$'s without this property is a long standing open problem that is closely related to the Davenport--Lewis--Schinzel problem (1959) on reducibility of separated polynomials. As opposed to previous work which mostly concerns indecomposable $f$, we answer both problems for decomposable $f=f_1\circ\cdots\circ f_r$, as long as the indecomposable factors $f_i\in\mathbb Q[x]$ …
Motivated by Euler's observation that the polynomial $x^{2} + x + 41$ takes on prime values for $0 \leq x \leq 39$, we search for large values of $x$ for which $N = x^{2} + x + 41$ is prime. To apply classical primality proving results based on the factorization of $N-1$, we choose $x$ to have the form $g(y)$, chosen so that $g(y)^{2} + g(y) + 40$ is reducible. Our main result is an explicit, 60,000 digit prime number of the form $x^{2} + x + 41$.
We prove sharp limit theorems on random walks on graphs with values in finite groups. We then apply these results (together with some elementary algebraic geometry, number theory, and representation theory) to finite quotients of lattices in semisimple Lie groups (specifically SL(n,Z) and Sp(2n, Z) to show that a ``random'' element in one of these lattices has irreducible characteristic polynomials (over the integers. The term ``random'' can be defined in at least two ways (in terms of height and also in terms of word length in terms of a generating set) -- we show the result using both defini…
We consider almost-primes of the form $f(p)$ where $f$ is an irreducible polynomial over $\mathbb Z$ and $p$ runs over primes. We improve a result of Richert for polynomials of degree at least $3$. In particular we show that, when the degree is large, there are infinitely many primes $p$ for which $f(p)$ has at most $°f+O(\log°f)$ prime factors.
The prime divisors of a polynomial $P$ with integer coefficients are those primes $p$ for which $P(x) \equiv 0 \pmod{p}$ is solvable. Our main result is that the common prime divisors of any several polynomials are exactly the prime divisors of some single polynomial. By combining this result with a theorem of Ax we get that for any system $F$ of multivariate polynomial equations with integer coefficients, the set of primes $p$ for which $F$ is solvable modulo $p$ is the set of prime divisors of some univariate polynomial. In addition, we prove results on the densities of the prime divisors of…
A second order polynomial sequence is of Fibonacci-type $\mathcal{F}_{n}$ (Lucas-type $\mathcal{L}_{n}$) if its Binet formula has a structure similar to that for Fibonacci (Lucas) numbers. Under certain conditions these polynomials are irreducible if and only if $n$ is a prime number. For example, the Fibonacci polynomials, Pell polynomials, Fermat polynomials, Lucas polynomials, Pell-Lucas polynomials, Fermat-Lucas polynomials are irreducible when $n$ is a prime number; and Chebyshev polynomials (second kind), Morgan-Voyce polynomials (Fibonacci type), and Vieta polynomials are reducible when…
A polynomial f(x) has emergent reducibility at depth n if f^{\circ k}(x) is irreducible for 0\leq k\leq n-1 but f^{\circ n}(x) is reducible. In this paper we prove that there are infinitely many irreducible cubics f \in \mathbb{Z}[x] with f\circ f reducible by exhibiting a one parameter family with this property.
We give necessary conditions satisfied by the set of odd prime divisors of binary perfect polynomials. This allows us to get a new characterization of all the known perfect binary polynomials.
We estimate the number of primes represented by a general quadratic polynomial with discriminant $Δ$, assuming that the corresponding real character is exceptional.
We improve Irving's method of the double-sieve by using the DHR sieve. By extending the upper and lower sieve functions into their respective non-elementary ranges, we are able to make improvements on the previous records on the number of prime factors of irreducible polynomials at prime arguments. In particular, we prove that irreducible quadratics over $\mathbb{Z}$ satisfying necessary local conditions are $P_4$ infinitely often.
We define an Artin prime for an integer $g$ to be a prime such that $g$ is a primitive root modulo that prime. Let $g\in \mathbb{Z}\setminus\{-1\}$ and not be a perfect square. A conjecture of Artin states that the set of Artin primes for $g$ has a positive density. In this paper we study a generalization of this conjecture for the primes produced by a polynomial and explore its connection with the problem of finding a fixed integer $g$ and a prime producing polynomial $f(x)$ with the property that a long string of consecutive primes produced by $f(x)$ are Artin primes for $g$. By employing so…
A distinguishing feature of certain intractable problems in prime number theory is the sparsity of the underlying sequence. Motivated by the general problem of finding primes in sparse polynomial sequences, we give an estimate for the number of primes of the shape $x^3 + 2y^3$ where $y$ is small.
An integral quadratic polynomial (with positive definite quadratic part) is called almost universal if it represents all but finitely many positive integers. In this paper, we introduce the conductor of a quadratic polynomial, and give an effective characterization of almost universal ternary quadratic polynomials with odd prime power conductor.
We get some results about the factorization of $ϕ_p(M) \in {\mathbb{F}}_2[x]$, where $p$ is a prime number, $ϕ_p$ is the corresponding cyclotomic polynomial and $M$ is a Mersenne prime (polynomial). By the way, we better understand the factorization of the sum of the divisors of $M^{2h}$, for a positive integer $h$.
A perfect polynomial over the binary field $\F_2$ is a polynomial $A \in \F_2[x]$ that equals the sum of all its divisors. If $\gcd(A,x^2-x) \neq 1$ then we call $A$ even. The list of all even perfect polynomials over $\F_2$ with at most 3 prime factors in known. The object of this paper is to give the list of all even perfect polynomials over $\F_2$ with four prime factors. These are all the known perfect polynomials with four prime factors over $\F_2$.
For a fixed quadratic irreducible polynomial $f$ with no fixed prime factors at prime arguments, we prove that there exist infinitely many primes $p$ such that $f(p)$ has at most 4 prime factors, improving a classical result of Richert who requires 5 in place of 4. Denoting by $P^+(n)$ the greatest prime factor of $n$, it is also proved that $P^+(f(p))>p^{0.847}$ infinitely often.
We establish, utilizing the Hardy-Littlewood Circle Method, an asymptotic formula for the number of pairs of primes whose differences lie in the image of a fixed polynomial. We also include a generalization of this result where differences are replaced with any integer linear combination of two primes.
In his paper Almost-Primes Represented by Quadratic Polynomials, Iwaniec proved that the polynomial n^2 + 1 takes on values with at most two prime factors (counted with multiplicity) infinitely often. He states that "in order to avoid technical complications, we shall restrict our proof to the polynomial n^2 + 1.". In this exposition, we follow Iwaniec's proof and show that for any irreducible quadratic polynomial G(n) (satisfying some obviously necessary hypotheses), G(n) has at most two prime factors for infinitely many values of n.
In the present paper we construct normal numbers in base $q$ by concatenating $q$-ary expansions of pseudo polynomials evaluated at the primes. This extends a recent result by Tichy and the author.
We discuss several enumerative results for irreducible polynomials of a given degree and pairs of relatively prime polynomials of given degrees in several variables over finite fields. Two notions of degree, the {\em total degree} and the {\em vector degree}, are considered. We show that the number of irreducibles can be computed recursively by degree and that the number of relatively prime pairs can be expressed in terms of the number of irreducibles. We also obtain asymptotic formulas for the number of irreducibles and the number of relatively prime pairs. The asymptotic formulas for the num…
Given a polynomial $ϕ$ over a global function field $K$ and a wandering base point $b\in K$, we give a geometric condition on $ϕ$ ensuring the existence of primitive prime divisors for almost all points in the orbit $\mathcal{O}_ϕ(b):=\{ϕ^n(b)\}_{n\geq0}$
The only (unitary) perfect polynomials over $\mathbb{F}_2$ that are products of $x$, $x+1$ and Mersenne primes are precisely the nine (resp. nine "classes") known ones. This follows from a new result about the factorization of $M^{2h+1} +1$, for a Mersenne prime $M$ and for a positive integer $h$. Other consequences of such a factorization are new results about odd perfect polynomials.
In this paper, we propose the use of Ramanujan class of polynomials for the construction of prime order elliptic curves using the CM-method. We compare (theoretically and experimentally) the efficiency of using this new class against the use of the Weber, $M_{D,l}(x)$ and $M_{D,p_1,p_2}(x)$ polynomials and show that they clearly outweigh all of them in the generation of prime order elliptic curves.
We establish asymptotic upper bounds on the number of zeros modulo $p$ of certain polynomials with integer coefficients, with $p$ prime numbers arbitrarily large. The polynomials we consider have degree of size $p$ and are obtained by truncating certain power series with rational coefficients that satisfy simple differential equations.
