On a problem of Diophantus for rationals

A positive integer $n$ is the area of a Heron triangle if and only if there is a non-zero rational number $\\tau$ such that the elliptic curve\n\\begin{equation*}\nE_{τ}^{(n)}: Y^{2} = X(X-nτ)(X+nτ^{-1})\n\\end{equation*}\nhas a rational point of order different than two. Such integers $n$ are called $\\tau$-congruent numbers. In this paper, we show that for a given positive integer $p$, and a given non-zero rational number $\\tau$, there exist infinitely many $\\tau$-congruent numbers in every residue class modulo $p$ whose corresponding elliptic curves have rank at least two.

It is known that a positive integer $n$ is the area of a right triangle with rational sides if and only if the elliptic curve $E^{(n)}: y^{2} = x(x^{2}-n^{2})$ has a rational point of order different than 2. A generalization of this result states that a positive integer $n$ is the area of a triangle with rational sides if and only if there is a nonzero rational number $\tau$ such that the elliptic curve $E^{(n)}_{\tau}: y^{2} = x(x-n\tau)(n+n\tau^{-1})$ has a rational point of order different than 2. Such $n$ are called $\tau$-congruent numbers. It is shown that for a given integer $m>1$, each congruence class modulo $m$ contains infinitely many distinct $\tau$-congruent numbers.

8 - THE SYSTEM OF RATIONAL NUMBERS

Simultaneous rational periodic points of degree-2 rational maps

Let $l$ be the prime $3$, $5$ or $7$, and let $m_{1}$,~$m_{2}$, $n_{1}$ and $n_{2}$ be non-zero rational numbers. We construct an infinite family of pairs of distinct quadratic fields $\mathbb{Q}(\sqrt{m_{1}D+n_{1}})$ and $\mathbb{Q}(\sqrt{m_{2}D+n_{2}})$ with $D\in\mathbb{Q}$ such that both class numbers are divisible by $l$, using rational points on an elliptic curve with positive Mordell-Weil rank to parametrize such quadratic fields.

Jordan triple maps

Irreducible polynomials and full elasticity in rings of integer-valued polynomials

A polynomial in one variable is an expression of the form a n x n + a n−1 x n−1 + ⋯ +a 0, where x is a variable*, where n is a nonnegative integer, and where a n , a n-1, ⋯ , a 0 are numbers. (Here, and in the remainder of the book, “number” will mean “rational number” unless otherwise stated.) The a’s are called the coefficients of the polynomial. The leading term of a polynomial is the term a i x i with a i ≠ 0 for which i is as large as possible. Normally one assumes a n ≠ 0, so a n x n is the leading term. The leading coefficient of a polynomial is the coefficient of the leading term; the degree is the exponent of x in the leading term. A polynomial of degree 0 is simply a nonzero rational number. The polynomial 0 has no leading term; it is considered to have degree −∞ in order to make the degree of the product of two polynomials equal to the sum of their degrees in all cases. For a similar reason, the leading coefficient of the polynomial 0 is considered to be 0. A polynomial is called monic if its leading coefficient is 1.

1. If R is a real closed field, then Artin [1] showed that every positive definite function f(Xl, . . ., Xj) in the rational function field R (X1,.. ., X.,) may be expressed as a sum of squares, thus solving a problem of Hilbert. However, Artin did not furnish the number of necessary squares. It was Pfister [8] who proved that 2' number of squares would suffice. Whether or not this upper bound is the best possible in general is still an outstanding open question, although for n < 2 this has been settled in the affirmative by Cassels-EllisonPfister [3]. To generalize this Hilbert problem with another ground field K instead of the real closed field R, Pfister asked in [8] whether 2n+2 might serve as an upper bound for K = Q, the rational number field. For n = 0, the theorem of Euler-Lagrange provides the answer. For n =1, Landau [5] showed that every positive definite polynomial in one variable over Q is a sum of eight squares of polynomials. Pourchet, however, recently proved [9] the rather startling fact that five is, in fact, the best possible bound in this case. In this short note we complete Pourchet's work by explicitly determining the best possible bound for K (X) where K is any formally real algebraic number field. We shall call this invariant the height. Thus, the reduced height m(F) of a field F is the least positive integer (or infinity) such that every sum of squares in F is a sum of m(F) number of squares. More precisely, we prove the following:

It follows from the work of Artin and Hooley that, under assumption of the generalised Riemann hypothesis, the density of the set of primes q for which a given non-zero rational number r is a primitive root modulo q can be written as an infinite product ∏p δp of local factors δp reflecting the degree of the splitting field of Xp - r at the primes p, multiplied by a somewhat complicated factor that corrects for the ‘entanglement’ of these splitting fields.We show how the correction factors arising in Artin's original primitive root problem and several of its generalisations can be interpreted as character sums describing the nature of the entanglement. The resulting description in terms of local contributions is so transparent that it greatly facilitates explicit computations, and naturally leads to non-vanishing criteria for the correction factors.The method not only applies in the setting of Galois representations of the multiplicative group underlying Artin's conjecture, but also in the GL2-setting arising for elliptic curves. As an application, we compute the density of the set of primes of cyclic reduction for Serre curves.

We use entropy rates and Schur concavity to prove that, for every integer k ≥2, every nonzero rational number q, and every real number α, the base-k expansions of α, q + α, and qα all have the same finite-state dimension and the same finite-state strong dimension. This extends, and gives a new proof of, Wall’s 1949 theorem stating that the sum or product of a nonzero rational number and a Borel normal number is always Borel normal.KeywordsShannon EntropyNonzero EntryNormal BaseNormal SequenceKolmogorov ComplexityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A nonzero rational integer has absolute value at least 1. A nonzero rational number has absolute value at least the inverse of any denominator. Liouville’s inequality (§ 3.5) is an extension of these estimates and provides a lower bound for the absolute value of any nonzero algebraic number. More specifically, if we are given finitely many (fixed) algebraic numbers γ l,...,γ t , and a polynomial P ∈ ℤ[X1,...,X t ] which does not vanish at the point (γ l,...,γ t ) then we can estimate from below |P(γ l,...,γ t )|. The lower bound will depend upon the degrees of P with respect to each of the X i ’s, the absolute values of its coefficients as well as some measure of the γ i ’s.KeywordsNumber FieldAlgebraic NumberMinimal PolynomialProduct FormulaAlgebraic IntegerThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Finite-state dimension and real arithmetic

Let C be a smooth genus one curve described by a quartic polynomial equation over the rational field \({{\mathbb {Q}}}\) with \(P\in C({{\mathbb {Q}}})\). We give an explicit criterion for the divisibility-by-2 of a rational point on the elliptic curve (C, P). This provides an analogue to the classical criterion of the divisibility-by-2 on elliptic curves described by Weierstrass equations. We employ this criterion to investigate the question of extending a rational D(q)-quadruple to a quintuple. We give concrete examples to which we can give an affirmative answer. One of these results implies that although the rational \( D(16t+9) \)-quadruple \(\{t, 16t+8,2 25t+14, 36t+20 \}\) can not be extended to a polynomial \( D(16t+9) \)-quintuple using a linear polynomial, there are infinitely many rational values of t for which the aforementioned rational \( D(16t+9) \)-quadruple can be extended to a rational \( D(16t+9) \)-quintuple. Moreover, these infinitely many values of t are parametrized by the rational points on a certain elliptic curve of positive Mordell–Weil rank.

Let p be a prime number. For each natural number n, we study the behavior of the function [Formula: see text] which enumerates the number of factorizations [Formula: see text] with [Formula: see text] a perfect square (mod p). The study of this function is inspired by the cognate function [Formula: see text] which enumerates the number of factorizations [Formula: see text] with [Formula: see text] a perfect square. The descent theory of elliptic curves would show that if [Formula: see text] is unbounded for squarefree values of n, then there are elliptic curves over the rational number field with arbitrarily large rank. In this note, we show for every prime p, [Formula: see text] is unbounded as n ranges over squarefree values, thus providing some evidence for the conjecture that [Formula: see text] is unbounded for squarefree n.