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A Heron triangle is a triangle that has three rational sides
$(a,b,c)$
and a rational area, whereas a perfect triangle is a Heron triangle that has three rational medians
$(k,l,m)$
. Finding a perfect triangle was stated as an open problem by Richard Guy [Unsolved Problems in Number Theory (Springer, New York, 1981)]. Heron triangles with two rational medians are parametrized by the eight curves
$C_{1},\ldots ,C_{8}$
mentioned in Buchholz and Rathbun [‘An infinite set of heron triangles with two rational medians’, Amer. Math. Monthly104(2) (1997), 106–115; ‘Heron triangles and elliptic curves’, Bull. Aust. Math.Soc.58 (1998), 411–421] and Bácskái et al. [Symmetries of triangles with two rational medians, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.65.6533, 2003]. In this paper, we reveal results on the curve
$C_{4}$
which has the property of satisfying conditions such that six of seven parameters given by three sides, two medians and area are rational. Our aim is to perform an extensive search to prove the nonexistence of a perfect triangle arising from this curve.
A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight
$k$
on
$\unicode[STIX]{x0393}_{0}(N)$
to a simpler function of
$k$
and
$N$
, showing that the two are equal whenever
$N$
is squarefree. We prove the converse of this theorem (with one small exception), thus providing a characterization of squarefree integers. We also establish a similar characterization of prime numbers in terms of the number of Hecke newforms of weight
$k$
on
$\unicode[STIX]{x0393}_{0}(N)$
.
It follows that a hypothetical fast algorithm for computing the number of such automorphic representations for even a single weight
$k$
would yield a fast test for whether
$N$
is squarefree. We also show how to obtain bounds on the possible square divisors of a number
$N$
that has been found not to be squarefree via this test, and we show how to probabilistically obtain the complete factorization of the squarefull part of
$N$
from the number of such automorphic representations for two different weights. If in addition we have the number of such Hecke newforms for even a single weight
$k$
, then we show how to probabilistically factor
$N$
entirely. All of these computations could be performed quickly in practice, given the number(s) of automorphic representations and modular forms as input.
We record
$\binom{42}{2}+\binom{23}{2}+\binom{13}{2}=1192$
functional identities that, apart from being amazingly amusing in themselves, find application in the derivation of Ramanujan-type formulas for
$1/\unicode[STIX]{x1D70B}$
and in the computation of mathematical constants.
We improve some previously known deterministic algorithms for finding integer solutions
$x,y$
to the exponential equation of the form
$af^{x}+bg^{y}=c$
over finite fields.
We use properties of the gamma function to estimate the products
$\prod _{k=1}^{n}(4k-3)/4k$
and
$\prod _{k=1}^{n}(4k-1)/4k$
, motivated by the work of Chen and Qi [‘Completely monotonic function associated with the gamma function and proof of Wallis’ inequality’, Tamkang J. Math.36(4) (2005), 303–307] and Mortici et al. [‘Completely monotonic functions and inequalities associated to some ratio of gamma function’, Appl. Math. Comput.240 (2014), 168–174].
In this paper, we prove some conjectures of K. Stolarsky concerning the first and third moments of the Beatty sequences with the golden section and its square.
Let
$G$
be a semisimple Lie group with associated symmetric space
$D$
, and let
$\unicode[STIX]{x1D6E4}\subset G$
be a cocompact arithmetic group. Let
$\mathscr{L}$
be a lattice inside a
$\mathbb{Z}\unicode[STIX]{x1D6E4}$
-module arising from a rational finite-dimensional complex representation of
$G$
. Bergeron and Venkatesh recently gave a precise conjecture about the growth of the order of the torsion subgroup
$H_{i}(\unicode[STIX]{x1D6E4}_{k};\mathscr{L})_{\operatorname{tors}}$
as
$\unicode[STIX]{x1D6E4}_{k}$
ranges over a tower of congruence subgroups of
$\unicode[STIX]{x1D6E4}$
. In particular, they conjectured that the ratio
$\log |H_{i}(\unicode[STIX]{x1D6E4}_{k};\mathscr{L})_{\operatorname{tors}}|/[\unicode[STIX]{x1D6E4}:\unicode[STIX]{x1D6E4}_{k}]$
should tend to a nonzero limit if and only if
$i=(\dim (D)-1)/2$
and
$G$
is a group of deficiency
$1$
. Furthermore, they gave a precise expression for the limit. In this paper, we investigate computationally the cohomology of several (non-cocompact) arithmetic groups, including
$\operatorname{GL}_{n}(\mathbb{Z})$
for
$n=3,4,5$
and
$\operatorname{GL}_{2}(\mathscr{O})$
for various rings of integers, and observe its growth as a function of level. In all cases where our dataset is sufficiently large, we observe excellent agreement with the same limit as in the predictions of Bergeron–Venkatesh. Our data also prompts us to make two new conjectures on the growth of torsion not covered by the Bergeron–Venkatesh conjecture.
For a finite field of odd cardinality
$q$
, we show that the sequence of iterates of
$aX^{2}+c$
, starting at
$0$
, always recurs after
$O(q/\text{log}\log q)$
steps. For
$X^{2}+1$
, the same is true for any starting value. We suggest that the traditional “birthday paradox” model is inappropriate for iterates of
$X^{3}+c$
, when
$q$
is 2 mod 3.
Robin’s criterion states that the Riemann hypothesis is true if and only if
$\unicode[STIX]{x1D70E}(n)<e^{\unicode[STIX]{x1D6FE}}n\log \log n$
for every positive integer
$n\geq 5041$
. In this paper we establish a new unconditional upper bound for the sum of divisors function, which improves the current best unconditional estimate given by Robin. For this purpose, we use a precise approximation for Chebyshev’s
$\unicode[STIX]{x1D717}$
-function.
This paper investigates interrelated price online inventory problems, in which decisions as to when and how much of a product to replenish must be made in an online fashion to meet some demand even without a concrete knowledge of future prices. The objective of the decision maker is to minimize the total cost while meeting the demands. Two different types of demand are considered carefully, that is, demands which are linearly and exponentially related to price. In this paper, the prices are online, with only the price range variation known in advance, and are interrelated with the preceding price. Two models of price correlation are investigated, namely, an exponential model and a logarithmic model. The corresponding algorithms of the problems are developed, and the competitive ratios of the algorithms are derived as the solutions by use of linear programming.
We generate ray-class fields over imaginary quadratic fields in terms of Siegel–Ramachandra invariants, which are an extension of a result of Schertz. By making use of quotients of Siegel–Ramachandra invariants we also construct ray-class invariants over imaginary quadratic fields whose minimal polynomials have relatively small coefficients, from which we are able to solve certain quadratic Diophantine equations.
Inspired by methods of N. P. Smart, we describe an algorithm to determine all Picard curves over
$\mathbb{Q}$
with good reduction away from 3, up to
$\mathbb{Q}$
-isomorphism. A correspondence between the isomorphism classes of such curves and certain quintic binary forms possessing a rational linear factor is established. An exhaustive list of integral models is determined and an application to a question of Ihara is discussed.
In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a particular substructure. A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding three-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid three-term geometric progressions. When unique factorization fails, or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets ‘greedily’, a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometric-progression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.
Consider two ordinary elliptic curves
$E,E^{\prime }$
defined over a finite field
$\mathbb{F}_{q}$
, and suppose that there exists an isogeny
$\unicode[STIX]{x1D713}$
between
$E$
and
$E^{\prime }$
. We propose an algorithm that determines
$\unicode[STIX]{x1D713}$
from the knowledge of
$E$
,
$E^{\prime }$
and of its degree
$r$
, by using the structure of the
$\ell$
-torsion of the curves (where
$\ell$
is a prime different from the characteristic
$p$
of the base field). Our approach is inspired by a previous algorithm due to Couveignes, which involved computations using the
$p$
-torsion on the curves. The most refined version of that algorithm, due to De Feo, has a complexity of
$\tilde{O} (r^{2})p^{O(1)}$
base field operations. On the other hand, the cost of our algorithm is
$\tilde{O} (r^{2})\log (q)^{O(1)}$
, for a large class of inputs; this makes it an interesting alternative for the medium- and large-characteristic cases.
In order to assess the security of cryptosystems based on the discrete logarithm problem in non-prime finite fields, as are the torus-based or pairing-based ones, we investigate thoroughly the case in
$\mathbb{F}_{p^{6}}$
with the number field sieve. We provide new insights, improvements, and comparisons between different methods to select polynomials intended for a sieve in dimension 3 using a special-
$\mathfrak{q}$
strategy. We also take into account the Galois action to increase the relation productivity of the sieving phase. To validate our results, we ran several experiments and real computations for various polynomial selection methods and field sizes with our publicly available implementation of the sieve in dimension 3, with special-
$\mathfrak{q}$
and various enumeration strategies.
We consider a smooth system of two homogeneous quadratic equations over
$\mathbb{Q}$
in
$n\geqslant 13$
variables. In this case, the Hasse principle is known to hold, thanks to the work of Mordell in 1959. The only local obstruction is over
$\mathbb{R}$
. In this paper, we give an explicit algorithm to decide whether a nonzero rational solution exists and, if so, compute one.
Let
$\mathbf{f}$
and
$\mathbf{g}$
be polynomials of a bounded Euclidean norm in the ring
$\mathbb{Z}[X]/\langle X^{n}+1\rangle$
. Given the polynomial
$[\mathbf{f}/\mathbf{g}]_{q}\in \mathbb{Z}_{q}[X]/\langle X^{n}+1\rangle$
, the NTRU problem is to find
$\mathbf{a},\mathbf{b}\in \mathbb{Z}[X]/\langle X^{n}+1\rangle$
with a small Euclidean norm such that
$[\mathbf{a}/\mathbf{b}]_{q}=[\mathbf{f}/\mathbf{g}]_{q}$
. We propose an algorithm to solve the NTRU problem, which runs in
$2^{O(\log ^{2}\unicode[STIX]{x1D706})}$
time when
$\Vert \mathbf{g}\Vert ,\Vert \mathbf{f}\Vert$
, and
$\Vert \mathbf{g}^{-1}\Vert$
are within some range. The main technique of our algorithm is the reduction of a problem on a field to one on a subfield. The GGH scheme, the first candidate of an (approximate) multilinear map, was recently found to be insecure by the Hu–Jia attack using low-level encodings of zero, but no polynomial-time attack was known without them. In the GGH scheme without low-level encodings of zero, our algorithm can be directly applied to attack this scheme if we have some top-level encodings of zero and a known pair of plaintext and ciphertext. Using our algorithm, we can construct a level-
$0$
encoding of zero and utilize it to attack a security ground of this scheme in the quasi-polynomial time of its security parameter using the parameters suggested by Garg, Gentry and Halevi [‘Candidate multilinear maps from ideal lattices’, Advances in cryptology — EUROCRYPT 2013 (Springer, 2013) 1–17].
We present a specialized point-counting algorithm for a class of elliptic curves over
$\mathbb{F}_{p^{2}}$
that includes reductions of quadratic
$\mathbb{Q}$
-curves modulo inert primes and, more generally, any elliptic curve over
$\mathbb{F}_{p^{2}}$
with a low-degree isogeny to its Galois conjugate curve. These curves have interesting cryptographic applications. Our algorithm is a variant of the Schoof–Elkies–Atkin (SEA) algorithm, but with a new, lower-degree endomorphism in place of Frobenius. While it has the same asymptotic asymptotic complexity as SEA, our algorithm is much faster in practice.
In this paper we describe how to compute smallest monic polynomials that define a given number field
$\mathbb{K}$
. We make use of the one-to-one correspondence between monic defining polynomials of
$\mathbb{K}$
and algebraic integers that generate
$\mathbb{K}$
. Thus, a smallest polynomial corresponds to a vector in the lattice of integers of
$\mathbb{K}$
and this vector is short in some sense. The main idea is to consider weighted coordinates for the vectors of the lattice of integers of
$\mathbb{K}$
. This allows us to find the desired polynomial by enumerating short vectors in these weighted lattices. In the context of the subexponential algorithm of Biasse and Fieker for computing class groups, this algorithm can be used as a precomputation step that speeds up the rest of the computation. It also widens the applicability of their faster conditional method, which requires a defining polynomial of small height, to a much larger set of number field descriptions.
We present JKL-ECM, an implementation of the elliptic curve method of integer factorization which uses certain twisted Hessian curves in a family studied by Jeon, Kim and Lee. This implementation takes advantage of torsion subgroup injection for families of elliptic curves over a quartic number field, in addition to the ‘small parameter’ speedup. We produced thousands of curves with torsion
$\mathbb{Z}/6\mathbb{Z}\oplus \mathbb{Z}/6\mathbb{Z}$
and small parameters in twisted Hessian form, which admit curve arithmetic that is ‘almost’ as fast as that of twisted Edwards form. This allows JKL-ECM to compete with GMP-ECM for finding large prime factors. Also, JKL-ECM, based on GMP, accepts integers of arbitrary size. We classify the torsion subgroups of Hessian curves over
$\mathbb{Q}$
and further examine torsion properties of the curves described by Jeon, Kim and Lee. In addition, the high-performance curves with torsion
$\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/8\mathbb{Z}$
of Bernstein et al. are completely recovered by the
$\mathbb{Z}/4\mathbb{Z}\oplus \mathbb{Z}/8\mathbb{Z}$
family of Jeon, Kim and Lee, and hundreds more curves are produced besides, all with small parameters and base points.