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We derive a q-supercongruence modulo the third power of a cyclotomic polynomial with the help of Guo and Zudilin’s method of creative microscoping [‘A q-microscope for supercongruences’, Adv. Math.346 (2019), 329–358] and the q-Dixon formula. As consequences, we give several supercongruences including
Recently, Lin and Liu [‘Congruences for the truncated Appell series
$F_3$
and
$F_4$
’, Integral Transforms Spec. Funct.31(1) (2020), 10–17] confirmed a supercongruence on the truncated Appell series
$F_3$
. Motivated by their work, we give a generalisation of this supercongruence by establishing a q-supercongruence modulo the fourth power of a cyclotomic polynomial.
When a page, represented by the interval
$[0,1]$
, is folded right over left
$n $
times, the right-hand fold contains a sequence of points. We specify these points using two different representation techniques, both involving binary signed-digit representations.
Let n and k be positive integers with
$n\ge k+1$
and let
$\{a_i\}_{i=1}^n$
be a strictly increasing sequence of positive integers. Let
$S_{n, k}:=\sum _{i=1}^{n-k} {1}/{\mathrm {lcm}(a_{i},a_{i+k})}$
. In 1978, Borwein [‘A sum of reciprocals of least common multiples’, Canad. Math. Bull.20 (1978), 117–118] confirmed a conjecture of Erdős by showing that
$S_{n,1}\le 1-{1}/{2^{n-1}}$
. Hong [‘A sharp upper bound for the sum of reciprocals of least common multiples’, Acta Math. Hungar.160 (2020), 360–375] improved Borwein’s upper bound to
$S_{n,1}\le {a_{1}}^{-1}(1-{1}/{2^{n-1}})$
and derived optimal upper bounds for
$S_{n,2}$
and
$S_{n,3}$
. In this paper, we present a sharp upper bound for
$S_{n,4}$
and characterise the sequences
$\{a_i\}_{i=1}^n$
for which the upper bound is attained.
We give a new q-analogue of the (A.2) supercongruence of Van Hamme. Our proof employs Andrews’ multiseries generalisation of Watson’s $_{8}\phi _{7}$ transformation, Andrews’ terminating q-analogue of Watson’s $_{3}F_{2}$ summation, a q-Watson-type summation due to Wei–Gong–Li and the creative microscoping method, developed by the author and Zudilin [‘A q-microscope for supercongruences’, Adv. Math.346 (2019), 329–358]. As a conclusion, we confirm a weaker form of Conjecture 4.5 by the author [‘Some generalizations of a supercongruence of van Hamme’, Integral Transforms Spec. Funct.28 (2017), 888–899].
Let
$a,b$
and n be positive integers and let
$S=\{x_1, \ldots , x_n\}$
be a set of n distinct positive integers. For
${x\in S}$
, define
$G_{S}(x)=\{d\in S: d<x, \,d\mid x \ \mathrm {and} \ (d\mid y\mid x, y\in S)\Rightarrow y\in \{d,x\}\}$
. Denote by
$[S^a]$
the
$n\times n$
matrix having the ath power of the least common multiple of
$x_i$
and
$x_j$
as its
$(i,j)$
-entry. We show that the bth power matrix
$[S^b]$
is divisible by the ath power matrix
$[S^a]$
if
$a\mid b$
and S is gcd closed (that is,
$\gcd (x_i, x_j)\in S$
for all integers i and j with
$1\le i, j\le n$
) and
$\max _{x\in S} \{|G_S (x)|\}=1$
. This confirms a conjecture of Shaofang Hong [‘Divisibility properties of power GCD matrices and power LCM matrices’, Linear Algebra Appl.428 (2008), 1001–1008].
Swisher [‘On the supercongruence conjectures of van Hamme’, Res. Math. Sci.2 (2015), Article no. 18] and He [‘Supercongruences on truncated hypergeometric series’, Results Math.72 (2017), 303–317] independently proved that Van Hamme’s (G.2) supercongruence holds modulo $p^4$ for any prime $p\equiv 1\pmod {4}$. Swisher also obtained an extension of Van Hamme’s (G.2) supercongruence for $p\equiv 3 \pmod 4$ and $p>3$. In this note, we give new one-parameter generalisations of Van Hamme’s (G.2) supercongruence modulo $p^3$ for any odd prime p. Our proof uses the method of ‘creative microscoping’ introduced by Guo and Zudilin [‘A q-microscope for supercongruences’, Adv. Math.346 (2019), 329–358].
In this paper, we discuss a connection between geometric measure theory and number theory. This method brings a new point of view for some number-theoretic problems concerning digit expansions. Among other results, we show that for each integer k, there is a number
$M>0$
such that if
$b_{1},\ldots ,b_{k}$
are multiplicatively independent integers greater than M, there are infinitely many integers whose base
$b_{1},b_{2},\ldots ,b_{k}$
expansions all do not have any zero digits.
Let p be a prime with
$p\equiv 1\pmod {4}$
. Gauss first proved that
$2$
is a quartic residue modulo p if and only if
$p=x^2+64y^2$
for some
$x,y\in \Bbb Z$
and various expressions for the quartic residue symbol
$(\frac {2}{p})_4$
are known. We give a new characterisation via a permutation, the sign of which is determined by
$(\frac {2}{p})_4$
. The permutation is induced by the rule
$x \mapsto y-x$
on the
$(p-1)/4$
solutions
$(x,y)$
to
$x^2+y^2\equiv 0 \pmod {p}$
satisfying
$1\leq x < y \leq (p-1)/2$
.
Following Bridgeman, we demonstrate several families of infinite dilogarithm identities associated with Fibonacci numbers, Lucas numbers, convergents of continued fractions of even periods, and terms arising from various recurrence relations.
Let
$k\geq 2$
be an integer. We prove that the 2-automatic sequence of odious numbers
$\mathcal {O}$
is a k-additive uniqueness set for multiplicative functions: if a multiplicative function f satisfies a multivariate Cauchy’s functional equation
$f(x_1+x_2+\cdots +x_k)=f(x_1)+f(x_2)+\cdots +f(x_k)$
for arbitrary
$x_1,\ldots ,x_k\in \mathcal {O}$
, then f is the identity function
$f(n)=n$
for all
$n\in \mathbb {N}$
.
The Thue–Morse sequence is a prototypical automatic sequence found in diverse areas of mathematics, and in computer science. We study occurrences of factors w within this sequence, or more precisely, the sequence of gaps between consecutive occurrences. This gap sequence is morphic; we prove that it is not automatic as soon as the length of w is at least
$2$
, thereby answering a question by J. Shallit in the affirmative. We give an explicit method to compute the discrepancy of the number of occurrences of the block
$\mathtt {01}$
in the Thue–Morse sequence. We prove that the sequence of discrepancies is the sequence of output sums of a certain base-
$2$
transducer.
When a page, represented by the interval
$[0,1],$
is folded right over left
$ n$
times, the right-hand fold contains a sequence of points. We specify these points and the order in which they appear in each fold. We also determine exactly where in the folded structure any point in
$[0,1]$
appears and, given any point on the bottom line of the structure, which point lies at each level above it.
We generalize the greedy and lazy $\beta $-transformations for a real base $\beta $ to the setting of alternate bases ${\boldsymbol {\beta }}=(\beta _0,\ldots ,\beta _{p-1})$, which were recently introduced by the first and second authors as a particular case of Cantor bases. As in the real base case, these new transformations, denoted $T_{{\boldsymbol {\beta }}}$ and $L_{{\boldsymbol {\beta }}}$ respectively, can be iterated in order to generate the digits of the greedy and lazy ${\boldsymbol {\beta }}$-expansions of real numbers. The aim of this paper is to describe the measure-theoretical dynamical behaviors of $T_{{\boldsymbol {\beta }}}$ and $L_{{\boldsymbol {\beta }}}$. We first prove the existence of a unique absolutely continuous (with respect to an extended Lebesgue measure, called the p-Lebesgue measure) $T_{{\boldsymbol {\beta }}}$-invariant measure. We then show that this unique measure is in fact equivalent to the p-Lebesgue measure and that the corresponding dynamical system is ergodic and has entropy $({1}/{p})\log (\beta _{p-1}\cdots \beta _0)$. We give an explicit expression of the density function of this invariant measure and compute the frequencies of letters in the greedy ${\boldsymbol {\beta }}$-expansions. The dynamical properties of $L_{{\boldsymbol {\beta }}}$ are obtained by showing that the lazy dynamical system is isomorphic to the greedy one. We also provide an isomorphism with a suitable extension of the $\beta $-shift. Finally, we show that the ${\boldsymbol {\beta }}$-expansions can be seen as $(\beta _{p-1}\cdots \beta _0)$-representations over general digit sets and we compare both frameworks.
We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f \in \mathbb {Z}[x]$. We use an explicit version of Mertens’ theorem for number fields to estimate a related sum over rational primes. For a given $f \in \mathbb {Z}[x]$, our result yields a finite list of primes that certifies the number of distinct irreducible factors of f.
We investigate, for given positive integers a and b, the least positive integer
$c=c(a,b)$
such that the quotient
$\varphi (c!\kern-1.2pt)/\varphi (a!\kern-1.2pt)\varphi (b!\kern-1.2pt)$
is an integer. We derive results on the limit of
$c(a,b)/(a+b)$
as a and b tend to infinity and show that
$c(a,b)>a+b$
for all pairs of positive integers
$(a,b)$
, with the exception of a set of density zero.
Merca [‘Congruence identities involving sums of odd divisors function’, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci.22(2) (2021), 119–125] posed three conjectures on congruences for specific convolutions of a sum of odd divisor functions with a generating function for generalised m-gonal numbers. Extending Merca’s work, we complete the proof of these conjectures.
We establish a family of q-supercongruences modulo the cube of a cyclotomic polynomial for truncated basic hypergeometric series. This confirms a weaker form of a conjecture of the present authors. Our proof employs a very-well-poised Karlsson–Minton type summation due to Gasper, together with the ‘creative microscoping’ method introduced by the first author in recent joint work with Zudilin.
The article focuses on the evaluation of convolution sums $${W_k}(n): = \mathop \sum \nolimits_{_{m < {n \over k}}} \sigma (m)\sigma (n - km)$$ involving the sum of divisor function $$\sigma (n)$$ for k =21, 33, and 35. In this article, our aim is to obtain certain Eisenstein series of level 21 and use them to evaluate the convolution sums for level 21. We also make use of the existing Eisenstein series identities for level 33 and 35 in evaluating the convolution sums for level 33 and 35. Most of the convolution sums were evaluated using the theory of modular forms, whereas we have devised a technique which is free from the theory of modular forms. As an application, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$, for (a, b)=(1, 7), (1, 11), (2, 3), and (2, 5).