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Summary: A triangular-square number is a number that is both triangular and square. We demonstrate how to generate all triangular-square numbers using only elementary geometric ideas. We also show how this method relates to the Pell equations x 2 − 2 y 2 = ± 1 . [ABSTRACT FROM AUTHOR]

Let P 1 , P 2 , ... , P l ∈ Z X , Y be homogeneous polynomials of degree 2, with nonpositive discriminant and let Q be a polynomial with integer coefficients. Under the assumption that 0 is a root of Q with odd multiplicity, we show that the variant of the Brocard–Ramanujan Diophantine equation Q n ! = ∏ i = 1 l P i X i , Y i α i has only finitely many integer solutions n , X ¯ l , Y ¯ l , α ¯ l , where X ¯ l = X 1 , X 2 , ... X l . [ABSTRACT FROM AUTHOR]

The Surányi–Hickerson conjecture is a long-standing unsolved problem of Diophantine equations. This conjecture states that all the solutions to ℓ 1 ! ⋯ ℓ m ! = k ! with k - ℓ m ≥ 2 are (ℓ 1 , ... , ℓ m ; k) = (6 , 7 ; 10) , (3 , 5 , 7 ; 10) , (2 , 5 , 14 ; 16) and (2, 3, 3, 7; 9). In this paper, we generalize the Surányi–Hickerson conjecture to ℓ 1 ! ⋯ ℓ m ! = k 1 ! ⋯ k n ! . We say that a solution (ℓ 1 , ... , ℓ m ; k 1 , ... , k n) is trivial if there exists a pair (i, j) such that | ℓ i - k j | = 1 . As in the Surányi–Hickerson conjecture, we give theoretical and computational results. In particular, we suggest that all non-trivial solutions to the equation ℓ 1 ! ℓ 2 = k 1 ! k 2 ! are (ℓ 1 , ℓ 2 ; k 1 , k 2) = (7 , 13 ; 4 , 15) , (14, 62; 7, 66) and (22, 54; 18, 57). [ABSTRACT FROM AUTHOR]

In this note we give exact formulas (and asymptotics) for the number of rational points of bounded height on weighted projective stacks over global function fields. [ABSTRACT FROM AUTHOR]

A Diophantine m-tuple over a finite field F q is a set { a 1 , ... , a m } of m distinct elements in F q ∗ such that a i a j + 1 is a square in F q whenever i ≠ j . In this paper, we study M(q), the maximum size of a Diophantine tuple over F q , assuming the characteristic of F q is fixed and q → ∞ . By explicit constructions, we improve the lower bound on M(q). In particular, this improves a recent result of Dujella and Kazalicki by a multiplicative factor. [ABSTRACT FROM AUTHOR]

Let { T n } n ≥ 0 be the sequence of Tribonacci numbers. In this paper, we study the exponential Diophantine equation T n - 2 x 3 y = c , for n , x , y ∈ Z ≥ 0 . In particular, we show that there is no integer c with at least six representations of the form T n - 2 x 3 y . [ABSTRACT FROM AUTHOR]

For an integer k ≥ 2 , let L (k) be the k–generalized Lucas sequence which starts with 0 , ... , 2 , 1 (a total of k terms) and for which each term afterwards is the sum of the k preceding terms. In this paper we assume that an integer c can be represented in at least two ways as the difference between a k–generalized Lucas number and a power of b, then using the theory of nonzero linear forms in logarithms of algebraic numbers, we bound all possible solutions on this representation of c in terms of b. Finally, combination our general result and some known reduction procedures based on the continued fraction algorithm, we find all the integers c and their representations for b ∈ [ 2 , 10 ] , this argument can be generalized to any b > 10 . [ABSTRACT FROM AUTHOR]

Let r ⩾ 3 r\geqslant 3 be an integer and 푄 any positive definite quadratic form in 푟 variables. We establish asymptotic formulae with power-saving error terms for the number of rational points of bounded height on singular hypersurfaces S Q S_{Q} defined by x 3 = Q ⁢ ( y 1 , … , y r ) ⁢ z x^{3}=Q(y_{1},\dots,y_{r})z . This confirms Manin’s conjecture for any S Q S_{Q} . Our proof is based on analytic methods, and uses some estimates for character sums and moments of 퐿-functions. In particular, one of the ingredients is Siegel’s mass formula in the argument for the case r = 3 r=3 . [ABSTRACT FROM AUTHOR]

We explore a generalization of the Markov numbers that is motivated by a specific generalized cluster algebra arising from an orbifold, in the sense of Chekhov and Shapiro. We give an explicit algorithm for computing these generalized Markov numbers and exhibit several patterns analogous to those that appear within the ordinary Markov numbers. Along the way, we present formulas related to continued fractions and snake graphs. [ABSTRACT FROM AUTHOR]

Fix a non-square integer k ≠ 0 . We show that the number of curves E B : y 2 = x 3 + k B 2 containing an integral point, where B ranges over positive integers less than N, is bounded by ≪ k N (log N) - 1 2 + ϵ . In particular, this implies that the number of positive integers B ≤ N such that - 3 k B 2 is the discriminant of an elliptic curve over Q is o(N). The proof involves a discriminant-lowering procedure on integral binary cubic forms. [ABSTRACT FROM AUTHOR]

Let S be a finite, fixed set of primes. In this paper, we show that the set of integers c which have at least two representations as a difference between a factorial and an S-unit is finite and effectively computable. In particular, we find all integers that can be written in at least two ways as a difference of a factorial and an S-unit associated with the set of primes { 2 , 3 , 5 , 7 } . [ABSTRACT FROM AUTHOR]

We give asymptotics for the number of isomorphism classes of Drinfeld F q [ T ] -modules over F q (T) of a given height, which satisfy prescribed sets of local conditions. This is done by relating our problem to a problem about counting points on weighted projective stacks. Our results for counting points of bounded height on weighted projective stacks over global function fields may be of independent interest. [ABSTRACT FROM AUTHOR]

Let { P n } n ≥ 0 be the sequence of Pell numbers defined by P 0 = 0 , P 1 = 1 and P n + 2 = 2 P n + 1 + P n for all n ≥ 0 and let { Q n } n ≥ 0 be its companion sequence, the Pell–Lucas numbers defined by Q 0 = Q 1 = 2 and Q n + 2 = 2 Q n + 1 + Q n for all n ≥ 0 . In this paper, we find all Pell and Pell–Lucas numbers which can be written as difference of two repdigits. It is shown that the largest Pell and Pell–Lucas numbers which can be written as difference of two repdigits are P 6 = 70 = 77 - 7 and Q 7 = 478 = 555 - 77. [ABSTRACT FROM AUTHOR]

The Manin–Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higher-dimensional smooth spherical Fano varieties.

Let f be an $L^2$ -normalized holomorphic newform of weight k on $\Gamma _0(N) \backslash \mathbb {H}$ with N squarefree or, more generally, on any hyperbolic surface $\Gamma \backslash \mathbb {H}$ attached to an Eichler order of squarefree level in an indefinite quaternion algebra over $\mathbb {Q}$ . Denote by V the hyperbolic volume of said surface. We prove the sup-norm estimate $$\begin{align*}\| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_{\varepsilon} (k V)^{\frac{1}{4}+\varepsilon} \end{align*}$$

Let us denote by Fn the n-th Fibonacci number. In this paper we show that for a fixed integer y there exists at most one integer exponent a > 0 such that the Diophantine equation Fn + Fm = ya has a solution (n, m, a) in positive integers satisfying n > m > 0, unless y = 2, 3, 4, 6 or 10. [ABSTRACT FROM AUTHOR]

Pythagorean triples can be represented by points of the unit circle with rational coordinates. These points form an abelian group, and we describe its structure. This structural description yields an enumeration of the normalized Pythagorean triples with a given hypotenuse. It also gives rise to an effective method for producing all such triples, and this effective method seems to be new. This paper is intended for the general mathematical audience, including undergraduate mathematics students. Therefore it contains background material, explanations, examples and drawings; and the proofs are complete. [ABSTRACT FROM AUTHOR]

We show that the system of equations $$ \begin{align*} \sum_{i=1}^{s} (x_{i}^{\,j}-y_{i}^{\,j}) = a_{j} \quad (1 \leqslant j \leqslant k) \end{align*} $$ has appreciably fewer solutions in the subcritical range $s than its homogeneous counterpart, provided that $a_{\ell } \neq 0$ for some $\ell \leqslant k-1$. Our methods use Vinogradov's mean value theorem in combination with a shifting argument. [ABSTRACT FROM AUTHOR]

The integer sequence defined by P n + 3 = P n + 1 + P n with initial conditions P 0 = 1 and P 1 = P 2 = 0 is known as the Padovan sequence P n n ∈ Z . A recurrence sequence { u n } n ∈ Z is said to be of Padovan-type if it satisfies the same recurrence relation as the Padovan sequence but with arbitrary initial values u 0 , u 1 , u 2 not all zero. The most famous Padovan-type sequence is the Perrin sequence given by u 0 = 3 , u 1 = 0 and u 2 = 2 . We show that every Padovan-type sequence has at most 2 zeros, except for nonzero multiples of shifts of the Padovan sequence which has 0-multiplicity 5. We also show that P n n ∈ Z has total multiplicity 62, i.e., there are 62 pairs (m , n) ∈ Z 2 with m < n for which P m = P n . As a consequence, we found that { P n } n ∈ Z has multiplicity 9, being 1 the most repeated Padovan number. Finally, we prove that the Perrin sequence has exactly 1 zero, total multiplicity 23 and multiplicity 4. [ABSTRACT FROM AUTHOR]

In this article we formalize in Mizar [1], [2] the final step of our attempt to formally construct a prime representing polynomial with 10 variables proposed by Yuri Matiyasevich in [4]. The first part of the article includes many auxiliary lemmas related to multivariate polynomials. We start from the properties of monomials, among them their evaluation as well as the power function on polynomials to define the substitution for multivariate polynomials. For simplicity, we assume that a polynomial and substituted ones as i-th variable have the same number of variables. Then we study the number of variables that are used in given multivariate polynomials. By the used variable we mean a variable that is raised at least once to a non-zero power. We consider both adding unused variables and eliminating them. The second part of the paper deals with the construction of the polynomial proposed by Yuri Matiyasevich. First, we introduce a diophantine polynomial over 4 variables that has roots in integers if and only if indicated variable is the square of a natural number, and another two is the square of an odd natural number. We modify the polynomial by adding two variables in such a way that the root additionally requires the divisibility of these added variables. Then we modify again the polynomial by adding two variables to also guarantee the nonnegativity condition of one of these variables. Finally, we combine the prime diophantine representation proved in [7] with the obtained polynomial constructing a prime representing polynomial with 10 variables. This work has been partially presented in [8] with the obtained polynomial constructing a prime representing polynomial with 10 variables in Theorem (85). [ABSTRACT FROM AUTHOR]

Let ℓ ⩾ 2 be a fixed positive integer and Q(y) be a positive definite quadratic form in ℓ variables with integral coefficients. The aim of this paper is to count rational points of bounded height on the cubic hypersurface defined by u3 = Q(y)z. We can get a power-saving result for a class of special quadratic forms and improve on some previous work. [ABSTRACT FROM AUTHOR]

Let d ≥ 2 be an integer which is not a square. We show that if (L n) n ≥ 0 is the Lucas sequence and (X m , Y m) m ≥ 1 is the mth solution of the Pell equation X 2 - d Y 2 = ± 1 , then the equation Y m = L n has at most two positive integer solutions (m, n) except for d = 2 when it has the three solutions (m , n) = (1 , 1) , (2 , 0) , (5 , 7) . [ABSTRACT FROM AUTHOR]

For any fixed nonzero integer h, we show that a positive proportion of integral binary quartic forms F do locally everywhere represent h, but do not globally represent h. We order classes of integral binary quartic forms by the two generators of their ring of ${\rm GL}_{2}({\mathbb Z})$ -invariants, classically denoted by I and J. [ABSTRACT FROM AUTHOR]

We study the large deviation behavior of lacunary sums (S n / n) n ∈ N with S n : = ∑ k = 1 n f (a k U) , n ∈ N , where U is uniformly distributed on [0, 1], (a k) k ∈ N is an Hadamard gap sequence, and f : R → R is a 1-periodic, (Lipschitz-)continuous mapping. In the case of large gaps, we show that the normalized partial sums satisfy a large deviation principle at speed n and with a good rate function which is the same as in the case of independent and identically distributed random variables U k , k ∈ N , having uniform distribution on [0, 1]. When the lacunary sequence (a k) k ∈ N is a geometric progression, then we also obtain large deviation principles at speed n, but with a good rate function that is different from the independent case, its form depending in a subtle way on the interplay between the function f and the arithmetic properties of the gap sequence. Our work generalizes some results recently obtained by Aistleitner, Gantert, Kabluchko, Prochno, and Ramanan [Large deviation principles for lacunary sums, preprint, 2020] who initiated this line of research for the case of lacunary trigonometric sums. [ABSTRACT FROM AUTHOR]

For integers k, we consider the affine cubic surface V k given by M (x) = x 1 2 + x 2 2 + x 3 2 - x 1 x 2 x 3 = k . We show that for almost all k the Hasse Principle holds, namely that V k (Z) is non-empty if V k (Z p) is non-empty for all primes p, and that there are infinitely many k's for which it fails. The Markoff morphisms act on V k (Z) with finitely many orbits and a numerical study points to some basic conjectures about these "class numbers" and Hasse failures. Some of the analysis may be extended to less special affine cubic surfaces. [ABSTRACT FROM AUTHOR]

Given a set of endomorphisms on P N , we establish an upper bound on the number of points of bounded height in the associated monoid orbits. Moreover, we give a more refined estimate with an associated lower bound when the monoid is free. Finally, we show that most sets of rational functions in one variable satisfy these more refined bounds. [ABSTRACT FROM AUTHOR]

We provide explicit formulae for primitive, integral solutions to the Diophantine equation x2+y2=M{x}^{2}+{y}^{2}=M, where MM is a product of powers of Pythagorean primes, i.e., of primes of the form 4n+14n+1. It turns out that this is a nice application of the theory of Gaussian integers.

Consider an odd prime number p≡2(mod3)p\equiv 2\hspace{0.3em}\left(\mathrm{mod}\hspace{0.3em}3). In this paper, the number of certain type of partitions of zero in Z/pZ{\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} is calculated using a combination of elementary combinatorics and number theory. The focus is on the three-part partitions of 0 in Z/pZ{\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} with all three parts chosen from the set of non-zero quadratic residues mod pp. Such partitions are divided into two types. Those with exactly two of the three parts identical are classified as type I. The type II partitions are those with all three parts being distinct. The number of partitions of each type is given. The problem of counting such partitions is well related to that of counting the number of non-trivial solutions to the Diophantine equation x2+y2+z2=0{x}^{2}+{y}^{2}+{z}^{2}=0 in the ring Z/pZ{\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}}. Correspondingly, solutions to this equation are also classified as type I or type II. We give the number of solutions to the equation corresponding to each type.