Courbes de Fermat et principe de Hasse

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.

On a problem of Diophantus for rationals

8 - THE SYSTEM OF RATIONAL NUMBERS

Simultaneous rational periodic points of degree-2 rational maps

Jordan triple maps

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

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.

PurposeLet p[1,r;t] be defined by ∑n=0∞p[1,r;t](n)qn=(E1Er)t, where t is a non-zero rational number, r ≥ 1 is an integer and Er=∏n=0∞(1−qr(n+1)) for |q| < 1. The function p[1,r;t](n) is the generalisation of the two-colour partition function p[1,r;−1](n). In this paper, the authors prove some new congruences modulo odd prime ℓ by taking r = 5, 7, 11 and 13, and non-integral rational values of t.Design/methodology/approachUsing q-series expansion/identities, the authors established general congruence modulo prime number for two-colour partition function.FindingsIn the paper, the authors study congruence properties of two-colour partition function for fractional values. The authors also give some particular cases as examples.Originality/valueThe partition functions for fractional value is studied in 2019 by Chan and Wang for Ramanujan's general partition function and then extended by Xia and Zhu in 2020. In 2021, Baruah and Das also proved some congruences related to fractional partition functions previously investigated by Chan and Wang. In this sequel, some congruences are proved for two-colour partitions in this paper. The results presented in the paper are original.

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.

Adelic quantum mechanics is form-invariant under an interchange of real andp-adic number fields as well as rings ofp-adic integers. We also show that in adelic quantum mechanics Feynman’s path integrals for quadratic actions with rational coefficients are invariant under changes of their entries within nonzero rational numbers.

The degrees-of-freedom of a K-user Gaussian interference channel (GIFC) has been defined to be the multiple of (1/2) log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> P at which the maximum sum of achievable rates grows with increasing P. In this paper, we establish that the degrees-of-freedom of three or more user, real, scalar GIFCs, viewed as a function of the channel coefficients, is discontinuous at points where all of the coefficients are non-zero rational numbers. More specifically, for all K > 2, we find a class of K-user GIFCs that is dense in the GIFC parameter space for which K/2 degrees-of-freedom are exactly achievable, and we show that the degrees-of-freedom for any GIFC with non-zero rational coefficients is strictly smaller than K/2. These results are proved using new connections with number theory and additive combinatorics.

Let g be a non-zero rational number. Let N_{g,t}(x) denote the number of primes p<=x for which the subgroup of the multiplicative group of the finite field having p elements that is generated by g mod p is of residual index t. In Part I, which is published in J. Number Theory 83 (2000), 155-181, a heuristic for N_{g,t}(x) was set up, under the assumption of the Generalized Riemann Hypothesis (GRH), and shown to be asymptotically exact. In this preprint we provide an alternative and rather shorter proof of this result.