Integers X1 Research Articles

Representation of the norm of ideals by quadratic forms with congruence conditions

Using a correspondence between the narrow ray class group modulo m of a quadratic field and a certain set of equivalence classes of binary quadratic forms proved by Furuta and Kubota, we find a quadratic form f and a pair of integers (x1,y1) such that the norm of all integral ideals a in a ray class is represented by f(mx+x1,my+y1) with some integers (x,y).

Recurrence with prescribed number of residues

In this paper we show that for every positive integer m there exist positive integers x1,x2,M such that the sequence (xn)n=1∞ defined by the Fibonacci recurrence xn+2=xn+1+xn, n=1,2,3,…, has exactly m distinct residues modulo M. As an application we show that for each integer m⩾2 there exists ξ∈R such that the sequence of fractional parts {ξφn}n=1∞, where φ=(1+5)/2, has exactly m limit points. Furthermore, we prove that for no real ξ≠0 it has exactly one limit point.

Subspace partitions of [formula omitted] containing direct sums

Let q be a prime power and n be a positive integer. A subspace partition of V=Fqn, the vector space of dimension n over Fq, is a collection Π of subspaces of V such that each nonzero vector of V is contained in exactly one subspace in Π; the multiset of dimensions of subspaces in Π is then called a Gaussian partition of V. We say that Πcontains a direct sum if there exist subspaces W1,…,Wk∈Π such that W1⊕⋯⊕Wk=V. In this paper, we study the problem of classifying the subspace partitions that contain a direct sum. In particular, given integers a1 and a2 with n>a1>a2≥1, our main theorem shows that if Π is a subspace partition of Fqn with mi subspaces of dimension ai for i=1,2, then Π contains a direct sum when a1x1+a2x2=n has a solution (x1,x2) for some integers x1,x2≥0 and m2 belongs to the union I of two natural intervals. The lower bound of I captures all subspace partitions with dimensions in {a1,a2} that are currently known to exist. Moreover, we show the existence of infinite classes of subspace partitions without a direct sum when m2∉I or when the condition on the existence of a nonnegative integral solution (x1,x2) is not satisfied. We further conjecture that this theorem can be extended to any number of distinct dimensions, where the number of subspaces in each dimension has appropriate bounds. These results offer further evidence of the natural combinatorial relationship between Gaussian and integer partitions (when q→1) as well as subspace and set partitions.

English

A metacyclic group H can be presented as 〈α,β: αn = 1, βm = αt, βαβ−1 = αr〉 for some n, m, t, r. Each endomorphism σ of H is determined by $$\sigma(\alpha)=\alpha^{x_1}\beta^{y_1}, \sigma(\beta)=\alpha^{x_2}\beta^{y_2}$$ for some integers x1, x2, y1, y2. We give sufficient and necessary conditions on x1, x2, y1, y2 for σ to be an automorphism.

Is there a computable upper bound for the height of a solution of a Diophantine equation with a unique solution in positive integers?

Abstract Let Bn = {xi · xj = xk : i, j, k ∈ {1, . . . , n}} ∪ {xi + 1 = xk : i, k ∈ {1, . . . , n}} denote the system of equations in the variables x1, . . . , xn. For a positive integer n, let _(n) denote the smallest positive integer b such that for each system of equations S ⊆ Bn with a unique solution in positive integers x1, . . . , xn, this solution belongs to [1, b]n. Let g(1) = 1, and let g(n + 1) = 22g(n) for every positive integer n. We conjecture that ξ (n) 6 g(2n) for every positive integer n. We prove: (1) the function ξ : N \ {0} → N \ {0} is computable in the limit; (2) if a function f : N \ {0} → N \ {0} has a single-fold Diophantine representation, then there exists a positive integer m such that f (n) < ξ (n) for every integer n > m; (3) the conjecture implies that there exists an algorithm which takes as input a Diophantine equation D(x1, . . . , xp) = 0 and returns a positive integer d with the following property: for every positive integers a1, . . . , ap, if the tuple (a1, . . . , ap) solely solves the equation D(x1, . . . , xp) = 0 in positive integers, then a1, . . . , ap 6 d; (4) the conjecture implies that if a set M ⊆ N has a single-fold Diophantine representation, then M is computable; (5) for every integer n > 9, the inequality ξ (n) < (22n−5 − 1)2n−5 + 1 implies that 22n−5 + 1 is composite.

A hypothetical upper bound on the heights of the solutions of a Diophantine equation with a finite number of solutions

Abstract We define a computable function f from positive integers to positive integers. We formulate a hypothesis which states that if a system S of equations of the forms xi· xj = xk and xi + 1 = xi has only finitely many solutions in non-negative integers x1, . . . , xi, then the solutions of S are bounded from above by f (2n). We prove the following: (1) the hypothesis implies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that the question of whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (3) the hypothesis implies that the question of whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (4) the hypothesis implies that if a set M ⊆ N has a finite-fold Diophantine representation, thenMis computable.

Conjecturally computable functions which unconditionally do not have any finite-fold Diophantine representation

We define functions f1,f2,g1,g2:N∖{0}→N∖{0} and prove that they do not have any finite-fold Diophantine representation. We conjecture that if a system S⊆{xi+xj=xk,xi⋅xj=xk:i,j,k∈{1,…,n}} has only finitely many solutions in positive integers x1,…,xn, then each such solution (x1,…,xn) satisfies x1,…,xn⩽22n−1. Assuming the conjecture, we prove: (1) the functions f1, f2, g1, g2 are computable, (2) there is an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions if the solution set is finite, (3) a finite-fold Diophantine representation of the function N∋n→2n∈N does not exist, (4) if a set M⊆N is recursively enumerable but not recursive, then a finite-fold Diophantine representation of M does not exist.

A conjecture on integer arithmetic which implies that there is an algorithm which to each Diophantine equation assigns an integer which is greater than the heights of integer (non-negative integer, rational) solutions, if these solutions form a finite set

We conjecture that if a system S ⊆ {xi + xj = xk, xi · xj = xk : i, j, k ∈ {1, . . . , n}} has only finitely many solutions in integers x1, . . . , xn, then each such solution (x1, . . . , xn) satisfies max (|x1|, . . . , |xn|) ≤ 22 . The conjecture implies that there is an algorithm which to each Diophantine equation assigns an integer which is greater than the heights of integer (non-negative integer, rational) solutions, if these solutions form a finite set. We describe an algorithm whose execution never terminates. If the conjecture is true, then the algorithm sequentially displays all integers n ≥ 2. If the conjecture is false, then the algorithm has a finite output which ends with an integer tuple (a1, . . . , an), where n ≥ 4, 22 n−1 < max (|a1|, . . . , |an|) ≤ 22 , and the system {xi + xj = xk : (i, j, k ∈ {1, . . . , n}) ∧ (ai + aj = ak)} ∪ {xi · xj = xk : (i, j, k ∈ {1, . . . , n}) ∧ (ai · aj = ak)} is a counterexample to the conjecture. The algorithm is implemented in MuPAD and Pascal. Mathematics Subject Classification: 03B30, 11U05

Does there Exist an Algorithm which to Each Diophantine Equation Assigns an Integer which is Greater than the Modulus of Integer Solutions, if these Solutions form a Finite Set?

Let En = {xi = 1; xi + xj = xk; xi · xj = xk : i; j; k ∈ {1,...,n}}. We conjecture that if a system $S \subseteq E_n$ has only finitely many solutions in integers x1,...,xn, then each such solution (x1,...,xn) satisfies |x1|,...,|xn| ≤ 22n−1. Assuming the conjecture, we prove: (1) there is an algorithm which to each Diophantine equation assigns an integer which is greater than the heights of integer (non-negative integer, rational) solutions, if these solutions form a finite set, (2) if a set $\cal{M} \subseteq \mathbb{N}$ is recursively enumerable but not recursive, then a finite-fold Diophantine representation of $\cal{M}$ does not exist.

On β-expansions of unity for rational and transcendental numbers β

Abstract We investigate the sequence of integers x 1, x 2, x 3, … lying in {0, 1, …, [β]} in a so-called Rényi β-expansion of unity 1 = $\sum\limits_{j = 1}^\infty {x_j \beta ^{ - j} } $ for rational and transcendental numbers β &gt; 1. In particular, we obtain an upper bound for two strings of consecutive zeros in the β-expansion of unity for rational β. For transcendental numbers β which are badly approximable by algebraic numbers of every large degree and bounded height, we obtain an upper bound for the Diophantine exponent of the sequence X = (xj)j=1∞ in terms of β.

Prime and composite integers close to powers of a number

We prove that, for any real numbers ξ ≠ 0 and ν, the sequence of integer parts [ξ2n + ν], n = 0, 1, 2, . . . , contains infinitely many composite numbers. Moreover, if the number ξ is irrational, then the above sequence contains infinitely many elements divisible by 2 or 3. The same holds for the sequence [ξ( − 2)n + νn], n = 0, 1, 2, . . . , where ν0, ν1, ν2, . . . all lie in a half open real interval of length 1/3. For this, we show that if a sequence of integers x1, x2, x3, . . . satisfies the recurrence relation xn+d = cxn + F(xn+1, . . . , xn+d-1) for each n ≥ 1, where c ≠ 0 is an integer, \({F(z_1,\dots,z_{d-1}) \in \mathbb {Z}[z_1,\dots,z_{d-1}],}\) and limn→ ∞|xn| = ∞, then the number |xn| is composite for infinitely many positive integers n. The proofs involve techniques from number theory, linear algebra, combinatorics on words and some kind of symbolic computation modulo 3.

On Two Exponents of Approximation Related to a Real Number and Its Square

AbstractFor each real number ξ, let denote the supremum of all real numbers λ such that, for each sufficiently large X, the inequalities |x0| ≤ X, |x0ξ – x1| ≤ X–λ and |x0ξ2 – x2| ≤ X–λ admit a solution in integers x0, x1 and x2 not all zero, and let denote the supremum of all real numbers ω such that, for each sufficiently large X, the dual inequalities |x0 + x1ξ + x2ξ2| ≤ X–ω, |x1| ≤ X and |x2| ≤ X admit a solution in integers x0, x1 and x2 not all zero. Answering a question of Y. Bugeaud and M. Laurent, we show that the exponent where ξ ranges through all real numbers with [ℚ(ξ):ℚ] &gt; 2 form a dense subset of the interval while, for the same values of ξ, the dual exponents form a dense subset of . Part of the proof rests on a result of V. Jarník showing that for any real number ξ with [ℚ(ξ):ℚ] &gt; 2.

THE CONVOLUTION SUM $\sum\limits_{m&lt;n/9}\sigma(m)\sigma(n-9m)$

The evaluation of the sum ∑m&lt;n/9σ(m)σ(n - 9m) is carried out for all positive integers n. This evaluation is used to detemine the number of solutions to [Formula: see text] in integers x1, x2, x3, x4, x5, x6, x7, x8.

Small solutions to linear congruences and Hecke equidistribution

The purpose of the present note is to give an explicit application of equidistribution of Hecke points to the problem of small solutions of linear congruences modulo primes. Let p be an odd prime. For a random system of d linear congruences in n variables modulo p, we shall ask how many “small” solutions it has, i.e. solutions in Z of size about pd/n, the natural limiting point. We will show that the answer has a limit distribution as p→∞. Questions of a similar flavour for varieties of higher degree have been studied, i.e. given an affine variety V ⊂ An over Z/pZ of codimension d, it may be regarded as defining a system of polynomial congruences for n integers (x1, . . . , xn) ∈ Zn. One can ask (if very optimistic) whether there always exists an integral solution so that max15i5n |xi| 5 Cpd/n, for some constant C depending only on the invariants of V , e.g. degree. This is easily seen to be false, but it turns out that one may prove weaker assertions of this nature: see, for example, [L]. The present note shows that in the seemingly easy case of linear varieties, the small solutions exhibit interesting statistical properties which are, somewhat surprisingly, related to automorphic forms. The question of small solutions of linear congruences is also closely related to the study of fractional parts of linear forms (cf. Section 7 below), which have been investigated by a number of authors using ergodic theory: see, e.g., Marklof [M1]. In our context, we shall use automorphic forms and spectral theory in place of ergodic theory; we are therefore able to give explicit error estimates. The application of the spectral theory of automorphic forms to certain Diophantine questions in number theory is not new; see, for instance, Sarnak’s ICM address [Sa2]. Hecke equidistribution in particular has been studied in great generality by Clozel, Oh and Ullmo in [COU]. Our problem involves passing from smooth to sharp cutoff test functions in the equidistribution results; for this we give a simple way of optimizing the harmonic

On the Relationship between the Multidimensional Dirichlet Divisor Problem and the Boundary of Zeros of ζ(s)F

where k is a positive integer, x ≥ 2, τk(n) is the number of solutions in positive integers x1 , . . . , xk of the equation x1 · · · xk = n , ζ(s) is the Riemann zeta-function, s = σ + it , i = −1, and σ = Re s , t = Im s . The multidimensional Dirichlet divisor problem consists in finding the correct order of growth of |Rk(x)| as x → +∞ for any k ≥ 2. Dirichlet posed this problem and proved that |Rk(x)| ≤ x1−1/k(c log x) , (1)

Small Solutions of ϕ1x12 + . . . + ϕnxn2 = 0

AbstractLet ϕ1, . . . , ϕn (n ≥ 2) be nonzero integers such that the equationis solvable in integers x1, . . . , xn not all zero. It is shown that there exists a solution satisfyingand that the constant 2 is best possible.

On Simultaneous Diagonal Inequalities

in integers x1, . . . , xs has been considered by a number of authors over the last quartercentury, starting with the work of Cook [9] and Pitman [13] on the case t = 2. More recently, Brudern and Cook [8] have shown that the above system is soluble provided that s is sufficiently large in terms of k and t and that the forms F1, . . . , Ft satisfy certain additional conditions. What has not yet been considered is the possibility of allowing the forms F1, . . . , Ft to have different degrees. However, with the recent work of Wooley [18], [20] on the corresponding problem for equations, the study of such systems has become a feasible prospect. In this paper we take a first step in that direction by studying the analogue of the system considered in [18] and [20]. Let λ1, . . . , λs and μ1, . . . , μs be real numbers such that for each i either λi or μi is nonzero. We define the forms F (x) = λ1x 3 1 + · · ·+ λsxs G(x) = μ1x 2 1 + · · ·+ μsxs

On the number of Solutions of a Diophantine equation of Frobenius

where n > 1 is a fixed positive integer and for which we have a fixed set of integers 0 < a1 < · · · < an satisfying (a1, ..., an) = 1. For any nonnegative integer N we say that a solution exists if there exist nonnegative integers x1, ..., xn satisfying equation (1.1). It is well known that for all sufficiently large N the equation has solutions. In fact for n = 2 the largest N for which no solution exists can be explicitly written as a1a2 − a1 − a2. However no such formula exists for n ≥ 3 as shown by Curtis in [2]. Several ingenious algorithms exist in the literature for calculating this bound which is known as the conductor. For example the reader may refer to [1, 8, 9, 10, 13] for several algorithms for calculating the conductor. The literature on this subject is vast and for different aspects of the problem we can mention for example [3, 4, 11, 12, 15].

Fast Deterministic Processor Allocation

Interval allocation has been suggested as a possible formalization for the PRAM of the (vaguely defined) processor allocation problem, which is of fundamental importance in parallel computing. The interval allocation problem is, given n nonnegative integers x1, ., xn, to allocate n nonoverlapping subarrays of sizes x1, ., xn from within a base array of O(Σnj=1xj) cells. We show that interval allocation problems of size n can be solved in O((log log n)3) time with optimal speedup on a deterministic CRCW PRAM. In addition to a general solution to the processor allocation problem, this implies an improved deterministic algorithm for the problem of approximate summation. For both interval allocation and approximate summation, the fastest previous deterministic algorithms have running times of Θ(log n/log log n). We describe an application to the problem of computing the connected components of an undirected graph. Finally we show that there is a nonuniform deterministic algorithm that solves interval allocation problems of size n in O(log log n) time with optimal speedup.

On the converse of Wolstenholme's Theorem

The problem of distinguishing prime numbers from composite numbers (. . .) is known to be one of the most important and useful in arithmetic. (. . .) The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated. Wilson’s Theorem states that if p is prime then (p− 1)! ≡ −1 (mod p). It is easy to see that the converse of Wilson’s Theorem also holds. Thus Wilson’s Theorem can be used to identify the primes. Another congruence identifying the primes is (p+ 1)(2p+ 1)(3p+ 1) . . . ((p− 1)p+ 1) ≡ 0 (mod (p− 1)!). (For a proof see [21].) It is not difficult to show that (2p−1 p−1 ) ≡ 1 (mod p) for all primes p. In 1819 Babbage [5, p. 271] observed that the stronger congruence (2p−1 p−1 ) ≡ 1 (mod p2) holds for all primes p ≥ 3, and Wolstenholme [5, p. 271], in 1862, proved that (2p−1 p−1 ) ≡ 1 (mod p3) for all primes p ≥ 5. The congruence (2n−1 n−1 ) ≡ 1 (mod n3) has no composite solutions n < 109. J. P. Jones ([9, problem B31, p. 47], [23, p. 21] and [12]) has conjectured that there are no composite solutions. Unlike that of Wilson’s Theorem the converse of Wolstenholme’s Theorem is a very difficult problem. A set S of positive integers is a Diophantine set if there exists a polynomial P (n, x1, . . . , xm) with integer coefficients such that n ∈ S if and only if there exist nonnegative integers x1, . . . , xm for which P (n, x1, . . . , xm) = 0. If we define Q(n, x1, . . . , xm) = n(1 − P (n, x1, . . . , xm)), then the set S is identical to the positive range of Q as n, x1, . . . , xm range over the nonnegative integers. One of the most important results obtained in the investigation of Hilbert’s tenth problem (which asks for an algorithm to decide whether a