Let p≥3 be a prime number. A Fermat curve over Q of exponent p is defined by an equation of the form axp+byp+czp=0, where a, b, c are non-zero rational numbers. We prove in this article that there exist infinitely many Fermat curves defined over Q, of exponent p, pairwise non Q-isomorphic, contradicting the Hasse principle.

Simultaneous rational periodic points of degree-2 rational maps

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.

A Thue equation is a Diophantine equation of the form f(x; y) = r, where f is an irreducible binary form of degree at least 3, and r is a given nonzero rational number. A set S of at least three positive integers is called a D13-set if the product of any of its three distinct elements is a perfect cube minus one. We prove that any D13-set is finite and, for any positive integer a, the two-tuple {a, 2a} is extendible to a D13-set 3-tuple, but not to a 4-tuple. Using the well-known Thue equation 2x3 - y3 = 1, we show that the only cubic-triangular number is 1.

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.

Abstract We consider some lacunary power series with rational coefficients in 𝕈p. We show that under certain conditions these series take transcendental values at non-zero rational number arguments, and we determine the classes of these transcendental values with respect to Mahler’s classification of p -adic numbers.

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli [Formula: see text] such that the numbers [Formula: see text], [Formula: see text] and [Formula: see text] are linearly dependent over [Formula: see text] for some positive integers [Formula: see text], must be of degree at most [Formula: see text]. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in [Formula: see text] defined over [Formula: see text]. On the other hand, we show that, with obvious exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to a hyperbola [Formula: see text], where [Formula: see text].

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.

A nonzero rational number is called a {\it cube sum} if it is of the form $a^3+b^3$ with $a,b\in{\Bbb Q}^\times$. In this paper, we prove that for any odd integer $k\geq 1$, there exist infinitely many cube-free odd integers $n$ with exactly $k$ distinct prime factors such that $2n$ is a cube sum (resp. not a cube sum). We present also a general construction of Heegner points and obtain an explicit Gross-Zagier formula which is used to prove the Birch and Swinnerton-Dyer conjecture for certain elliptic curves related to the cube sum problem.

Rational products of singular moduli

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.

Two rank \(n\), integral quadratic forms \(f\) and \(g\) are said projectively equivalent if there exist nonzero rational numbers \(r\) and \(s\) such that \(rf\) and \(sg\) are rationally equivalent. Two odd dimensional, integral quadratic forms \(f\) and \(g\) are projectivelly equivalent if and only if their adjoints are rationally equivalent. We prove that a canonical representative of each projective class of forms of odd rank, exists and is unique up to genus (integral equivalence for indefinite forms). We give a useful characterization of this canonical representative. An explicit construction of integral classes with square-free determinant is given. As a consequence, two tables of ternary and quinary integral quadratic forms of index \(1\) and with square-free determinant are presented.

Abstract 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.

Given a multiplicative group of nonzero rational numbers and a positive integer m m , we consider the problem of determining the density of the set of primes p p for which the order of the reduction modulo p p of the group is divisible by m m . In the case when the group is finitely generated the density is explicitly computed. Some examples of groups with infinite rank are considered.

An algorithm for computing mixed sums of products of Bernoulli polynomials and Euler polynomials

We introduce two notions of equivalence for rational quadratic forms. Two $n$-ary rational quadratic forms are commensurable if they possess commensurable groups of automorphisms up to isometry. Two $n$-ary rational quadratic forms $F$ and $G$ are projectivelly equivalent if there are nonzero rational numbers $r$ and $s$ such that $rF$ and $sG$ are rationally equivalent. It is shown that if $F$\ and $G$\ have Sylvester signature $\{-,+,+,...,+\}$ then $F$\ and $G$\ are commensurable if and only if they are projectivelly equivalent. The main objective of this paper is to obtain a complete system of (computable) numerical invariants of rational $n$-ary quadratic forms up to projective equivalence. These invariants are a variation of Conway's $p$-excesses. Here the cases $n$ odd and $n$ even are surprisingly different. The paper ends with some examples

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For each nonzero rational number r, in [1], we considered the problem of approximating G(r) with partial sums of the series (1.1). In the case that an ≡ 1 and s = 1, we asked how well one can approximate e by the partial sums ∑n `=0 1 `! . J. Sondow [6] conjectured that exactly two of these partial sums are also convergents to the continued fraction of e. Among several results, Sondow and K. Schalm [7], proved that for almost all positive integers n, the partial sum ∑n `=0 1 `! is not a convergent to the continued fraction of e. Thus, the probability of obtaining a convergent to the continued fraction of e upon randomly choosing one of the first n partial sums of the power series of e tends to zero as n → ∞. Knowledge of the continued fraction of e, where a is a nonzero integer, and the best possible diophantine approximation of e, discovered by S. Ramanujan [5] and rediscovered by C. S. Davis [4], enabled the authors to prove in [2] that at most Oa(logM) of the first M convergents to the continued fraction of e are also partial sums of the corresponding power series. In [1], we considered general hypergeometric functions pFp(a1, . . . , ap; b1, . . . , bp; r) and showed that among their first N partial sums, no more than O(logN) are convergents to the continued fraction of pFp(a1, . . . , ap; b1, . . . , bp; r). Observe that this result includes e as a special case and that this particular corollary is a dual of the aforementioned result established in [2]. Moreover, when an is a real Dirichlet character or when {an}, n > 0, are the coefficients of an L-series attached to an elliptic curve without complex multiplication, we proved similar theorems in [1]. Lastly, we remark that in [2], we, in fact, proved Sondow’s conjecture. At the focal point of our study in [1] and in this paper are the following two definitions. For any rational number μ = a/b, with (a, b) = 1, consider the height H(μ) of μ, given by H(μ) = max{|a|, |b|}. For any real number α, and any positive real number δ, denote

In this article, we give a new lower bound for the dimension of the linear space over the rationals spanned by 1 and values of polylogarithmic functions at a non-zero rational number. Our proof uses Padé approximation following the argument of T. Rivoal, however we adapt a new linear independence criterion due to S. Fischler and W. Zudilin. We also present an example of the linear space of dimension $\\geqslant 3$ over $\\mathbf{Q}$, which is generated by 1 and polylogarithms.

On a problem of Diophantus for rationals