In this paper we show that a polynomial equation admits infinitely many prime-tuple solutions, assuming only that the equation satisfies suitable local conditions and the polynomial is sufficiently non-degenerate algebraically. Our notion of algebraic non-degeneracy is related to the $h$-invariant introduced by W. M. Schmidt. Our results prove a conjecture by B. Cook and Á. Magyar for hypersurfaces of degree 3.

Let p(n) denote the partition function. In this paper, we will prove that for $n\ges 222$,

$$\left| {\matrix{ {p(n)} & {p(n + 1)} & {p(n + 2)} \cr {p(n-1)} & {p(n)} & {p(n + 1)} \cr {p(n-2)} & {p(n-1)} & {p(n)} \cr } } \right| > 0.{\rm }$$

$n\ges 222$

$$(p(n)^2-p(n-1)p(n+1))^2-(p(n-1)^2-p(n-2)p(n))(p(n+1)^2-p(n)p(n+2))>0.$$

In 2010, Hei-Chi Chan introduced the cubic partition function a(n) in connection with Ramanujan's cubic continued fraction. Chen and Lin, and Ahmed, Baruah and Dastidar proved that a(25n + 22) ≡ 0 (mod 5) for n ⩾ 0. In this paper, we prove several infinite families of congruences modulo 5 and 7 for a(n). Our results generalize the congruence a(25n + 22) ≡ 0 (mod 5) and four congruences modulo 7 for a(n) due to Chen and Lin. Moreover, we present some non-standard congruences modulo 5 for a(n) by using an identity of Newman. For example, we prove that $a((({15\times 17^{3\alpha }+1})/{8})) \equiv 3^{\alpha +1} \ ({\rm mod}\ 5)$ for α ⩾ 0.

We prove that, for any finite set $A\subset \mathbb{Q}$ with $|AA|\leqslant K|A|$ and any positive integer $k$, the $k$-fold product set of the shift $A+1$ satisfies the bound

$$\begin{eqnarray}|\{(a_{1}+1)(a_{2}+1)\cdots (a_{k}+1):a_{i}\in A\}|\geqslant \frac{|A|^{k}}{(8k^{4})^{kK}}.\end{eqnarray}$$

$K$

$c\log |A|$

$c=c(k)$

$\unicode[STIX]{x1D6EC}$

Let C be a set of positive integers. In this paper, we obtain an algorithm for computing all subsets A of positive integers which are minimals with the condition that if x1 + … + xn is a partition of an element in C, then at least a summand of this partition belongs to A. We use techniques of numerical semigroups to solve this problem because it is equivalent to give an algorithm that allows us to compute all the numerical semigroups which are maximals with the condition that has an empty intersection with the set C.

A subset $A$ of a finite abelian group $G$ is called $(k,l)$-sum-free if the sum of $k$ (not necessarily distinct) elements of $A$ never equals the sum of $l$ (not necessarily distinct) elements of $A$. We find an explicit formula for the maximum size of a $(k,l)$-sum-free subset in $G$ for all $k$ and $l$ in the case when $G$ is cyclic by proving that it suffices to consider $(k,l)$-sum-free intervals in subgroups of $G$. This simplifies and extends earlier results by Hamidoune and Plagne [‘A new critical pair theorem applied to sum-free sets in abelian groups’, Comment. Math. Helv. 79(1) (2004), 183–207] and Bajnok [‘On the maximum size of a $(k,l)$-sum-free subset of an abelian group’, Int. J. Number Theory 5(6) (2009), 953–971].

We give the generating function of split $(n+t)$-colour partitions and obtain an analogue of Euler’s identity for split $n$-colour partitions. We derive a combinatorial relation between the number of restricted split $n$-colour partitions and the function $\unicode[STIX]{x1D70E}_{k}(\unicode[STIX]{x1D707})=\sum _{d|\unicode[STIX]{x1D707}}d^{k}$. We introduce a new class of split perfect partitions with $d(a)$ copies of each part $a$ and extend the work of Agarwal and Subbarao [‘Some properties of perfect partitions’, Indian J. Pure Appl. Math 22(9) (1991), 737–743].

Let $p\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}4$ be a prime number. We use a number field variant of Vinogradov’s method to prove density results about the following four arithmetic invariants: (i) $16$-rank of the class group $\text{Cl}(-4p)$ of the imaginary quadratic number field $\mathbb{Q}(\sqrt{-4p})$; (ii) $8$-rank of the ordinary class group $\text{Cl}(8p)$ of the real quadratic field $\mathbb{Q}(\sqrt{8p})$; (iii) the solvability of the negative Pell equation $x^{2}-2py^{2}=-1$ over the integers; (iv) $2$-part of the Tate–Šafarevič group $\unicode[STIX]{x0428}(E_{p})$ of the congruent number elliptic curve $E_{p}:y^{2}=x^{3}-p^{2}x$. Our results are conditional on a standard conjecture about short character sums.

We consider the function $f(n)$ that enumerates partitions of weight $n$ wherein each part appears an odd number of times. Chern [‘Unlimited parity alternating partitions’, Quaest. Math. (to appear)] noted that such partitions can be placed in one-to-one correspondence with the partitions of $n$ which he calls unlimited parity alternating partitions with smallest part odd. Our goal is to study the parity of $f(n)$ in detail. In particular, we prove a characterisation of $f(2n)$ modulo 2 which implies that there are infinitely many Ramanujan-like congruences modulo 2 satisfied by the function $f.$ The proof techniques are elementary and involve classical generating function dissection tools.

Let $\mathbf{f}=(f_{1},\ldots ,f_{R})$ be a system of polynomials with integer coefficients in which the degrees need not all be the same. We provide sufficient conditions for which the system of equations $f_{j}(x_{1},\ldots ,x_{n})=0~(1\leqslant j\leqslant R)$ satisfies a general local to global type statement, and has a solution where each coordinate is prime. In fact we obtain the asymptotic formula for number of such solutions, counted with a logarithmic weight, under these conditions. We prove the statement via the Hardy–Littlewood circle method. This is a generalization of the work of Cook and Magyar [‘Diophantine equations in the primes’, Invent. Math.198 (2014), 701–737], where they obtained the result when the polynomials of $\mathbf{f}$ all have the same degree. Hitherto, results of this type for systems of polynomial equations involving different degrees have been restricted to the diagonal case.

In 2005, using a famous lemma of Atkin and Swinnerton-Dyer (Some properties of partitions, Proc. Lond. Math. Soc. (3) 4 (1954), 84–106), Yesilyurt (Four identities related to third order mock theta functions in Ramanujan’s lost notebook, Adv. Math. 190 (2005), 278–299) proved four identities for third order mock theta functions found on pages 2 and 17 in Ramanujan’s lost notebook. The primary purpose of this paper is to offer new proofs in the spirit of what Ramanujan might have given in the hope that a better understanding of the identities might be gained. Third order mock theta functions are intimately connected with ranks of partitions. We prove new dissections for two rank generating functions, which are keys to our proof of the fourth, and the most difficult, of Ramanujan’s identities. In the last section of this paper, we establish new relations for ranks arising from our dissections of rank generating functions.

We study almost prime solutions of systems of Diophantine equations in the Birch setting. Previous work shows that there exist integer solutions of size $B$ with each component having no prime divisors below $B^{1/u}$, where $u$ equals $c_{0}n^{3/2}$, $n$ is the number of variables and $c_{0}$ is a constant depending on the degree and the number of equations. We improve the polynomial growth $n^{3/2}$ to the logarithmic $(\log n)(\log \log n)^{-1}$. Our main new ingredients are the generalization of the Brüdern–Fouvry vector sieve in any dimension and the incorporation of smooth weights into the Davenport–Birch version of the circle method.

Assuming a conjecture on distinct zeros of Dirichlet $L$-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good error terms gives information on the location of zeros of $L$-functions. Similar results are obtained for an integer in a congruence class expressed as the sum of two primes.

We compute the limit shape for several classes of restricted integer partitions, where the restrictions are placed on the part sizes rather than the multiplicities. Our approach utilizes certain classes of bijections which map limit shapes continuously in the plane. We start with bijections outlined in [43], and extend them to include limit shapes with different scaling functions.

An additive basis $A$ is finitely stable when the order of $A$ is equal to the order of $A\cup F$ for all finite subsets $F\subseteq \mathbb{N}$. We give a sufficient condition for an additive basis to be finitely stable. In particular, we prove that $\mathbb{N}^{2}$ is finitely stable.

We study a combinatorial problem that recently arose in the context of shape optimization: among all triangles with vertices $(0,0)$, $(x,0)$, and $(0,y)$ and fixed area, which one encloses the most lattice points from $\mathbb{Z}_{{>}0}^{2}$? Moreover, does its shape necessarily converge to the isosceles triangle $(x=y)$ as the area becomes large? Laugesen and Liu suggested that, in contrast to similar problems, there might not be a limiting shape. We prove that the limiting set is indeed non-trivial and contains infinitely many elements. We also show that there exist “bad” areas where no triangle is particularly good at capturing lattice points and show that there exists an infinite set of slopes $y/x$ such that any associated triangle captures more lattice points than any other fixed triangle for infinitely many (and arbitrarily large) areas; this set of slopes is a fractal subset of $[1/3,3]$ and has Minkowski dimension of at most $3/4$.

We generalize Skriganov’s notion of weak admissibility for lattices to include standard lattices occurring in Diophantine approximation and algebraic number theory, and we prove estimates for the number of lattice points in sets such as aligned boxes. Our result improves on Skriganov’s celebrated counting result if the box is sufficiently distorted, the lattice is not admissible, and, e.g., symplectic or orthogonal. We establish a criterion under which our error term is sharp, and we provide examples in dimensions $2$ and $3$ using continued fractions. We also establish a similar counting result for primitive lattice points, and apply the latter to the classical problem of Diophantine approximation with primitive points as studied by Chalk, Erdős, and others. Finally, we use o-minimality to describe large classes of sets to which our counting results apply.

We study the Goldbach problem for primes represented by the polynomial $x^{2}+y^{2}+1$. The set of such primes is sparse in the set of all primes, but the infinitude of such primes was established by Linnik. We prove that almost all even integers $n$ satisfying certain necessary local conditions are representable as the sum of two primes of the form $x^{2}+y^{2}+1$. This improves a result of Matomäki, which tells us that almost all even $n$ satisfying a local condition are the sum of one prime of the form $x^{2}+y^{2}+1$ and one generic prime. We also solve the analogous ternary Goldbach problem, stating that every large odd $n$ is the sum of three primes represented by our polynomial. As a byproduct of the proof, we show that the primes of the form $x^{2}+y^{2}+1$ contain infinitely many three-term arithmetic progressions, and that the numbers $\unicode[STIX]{x1D6FC}p~(\text{mod}~1)$, with $\unicode[STIX]{x1D6FC}$ irrational and $p$ running through primes of the form $x^{2}+y^{2}+1$, are distributed rather uniformly.

Let $s\geqslant 3$ be a fixed positive integer and let $a_{1},\ldots ,a_{s}\in \mathbb{Z}$ be arbitrary. We show that, on average over $k$, the density of numbers represented by the degree $k$ diagonal form

$$\begin{eqnarray}a_{1}x_{1}^{k}+\cdots +a_{s}x_{s}^{k}\end{eqnarray}$$

$k$

Recently, Freiman et al. [‘Small doubling in ordered groups’, J. Aust. Math. Soc. 96(3) (2014), 316–325] proved two ‘structure theorems’ for ordered groups. We give elementary proofs of these two theorems.