In his 1984 AMS Memoir, Andrews introduced the family of functions $c\phi_k(n)$, the number of k-coloured generalized Frobenius partitions of n. In 2019, Chan, Wang and Yang systematically studied the arithmetic properties of $\textrm{C}\Phi_k(q)$ for $2\leq k\leq17$ by utilizing the theory of modular forms, where $\textrm{C}\Phi_k(q)$ denotes the generating function of $c\phi_k(n)$. In this paper, we first establish another expression of $\textrm{C}\Phi_{12}(q)$ with integer coefficients, then prove some congruences modulo small powers of 3 for $c\phi_{12}(n)$ by using some parameterized identities of theta functions due to A. Alaca, S. Alaca and Williams. Finally, we conjecture three families of congruences modulo powers of 3 satisfied by $c\phi_{12}(n)$.

We prove the following conjecture of Z.-W. Sun [‘On congruences related to central binomial coefficients’, J. Number Theory 13(11) (2011), 2219–2238]. Let p be an odd prime. Then

$$ \begin{align*} \sum_{k=1}^{p-1}\frac{\binom{2k}k}{k2^k}\equiv-\frac12H_{{(p-1)}/2}+\frac7{16}p^2B_{p-3}\pmod{p^3}, \end{align*} $$

where $H_n$ is the nth harmonic number and $B_n$ is the nth Bernoulli number. In addition, we evaluate $\sum _{k=0}^{p-1}(ak+b)\binom {2k}k/2^k$ modulo $p^3$ for any p-adic integers $a, b$.

In this paper, we mainly prove the following conjectures of Sun [16]: Let p > 3 be a prime. Then

\begin{align*}&A_{2p}\equiv A_2-\frac{1648}3p^3B_{p-3}\ ({\rm{mod}}\ p^4),\\&A_{2p-1}\equiv A_1+\frac{16p^3}3B_{p-3}\ ({\rm{mod}}\ p^4),\\&A_{3p}\equiv A_3-36738p^3B_{p-3}\ ({\rm{mod}}\ p^4),\end{align*}

where $A_n=\sum_{k=0}^n\binom{n}k^2\binom{n+k}{k}^2$ is the nth Apéry number, and Bn is the nth Bernoulli number.

Ranks of partitions play an important role in the theory of partitions. They provide combinatorial interpretations for Ramanujan’s famous congruences for partition functions. We establish a family of congruences modulo powers of $5$ for ranks of partitions.

We investigate Eisenstein congruences between the so-called Euler systems of Garrett–Rankin–Selberg type. This includes the cohomology classes of Beilinson–Kato, Beilinson–Flach, and diagonal cycles. The proofs crucially rely on different known versions of the Bloch–Kato conjecture, and are based on the study of the Perrin-Riou formalism and the comparison between the different p-adic L-functions.

In 2019, Andrews and Newman [‘Partitions and the minimal excludant’, Ann. Comb. 23(2) (2019), 249–254] introduced the arithmetic function $\sigma \textrm {mex}(n)$, which denotes the sum of minimal excludants over all the partitions of n. Baruah et al. [‘A refinement of a result of Andrews and Newman on the sum of minimal excludants’, Ramanujan J., to appear] showed that the sum of minimal excludants over all the partitions of n is the same as the number of partition pairs of n into distinct parts. They proved three congruences modulo $4$ and $8$ for two functions appearing in this refinement and conjectured two further congruences modulo $8$ and $16$. We confirm these two conjectures by using q-series manipulations and modular forms.

In this paper, we mainly prove the following conjectures of Z.-W. Sun (J. Number Theory 133 (2013), 2914–2928): let $p>2$ be a prime. If $p=x^2+3y^2$ with $x,y\in \mathbb {Z}$ and $x\equiv 1\ ({\rm {mod}}\ 3)$, then

\[ x\equiv\frac14\sum_{k=0}^{p-1}(3k+4)\frac{f_k}{2^k}\equiv\frac12\sum_{k=0}^{p-1}(3k+2)\frac{f_k}{({-}4)^k}\ ({\rm{mod}}\ p^2), \]

$p\equiv 1\pmod 3$

\[ \sum_{k=0}^{p-1}\frac{f_k}{2^k}\equiv\sum_{k=0}^{p-1}\frac{f_k}{({-}4)^k}\ ({\rm{mod}}\ p^3), \]

$f_n=\sum _{k=0}^n\binom {n}k^3$

$n$

Let $Q(n)$ denote the number of partitions of n into distinct parts. Merca [‘Ramanujan-type congruences modulo 4 for partitions into distinct parts’, An. Şt. Univ. Ovidius Constanţa 30(3) (2022), 185–199] derived some congruences modulo $4$ and $8$ for $Q(n)$ and posed a conjecture on congruences modulo powers of $2$ enjoyed by $Q(n)$. We present an approach which can be used to prove a family of internal congruence relations modulo powers of $2$ concerning $Q(n)$. As an immediate consequence, we not only prove Merca’s conjecture, but also derive many internal congruences modulo powers of $2$ satisfied by $Q(n)$. Moreover, we establish an infinite family of congruence relations modulo $4$ for $Q(n)$.

Recently, when studying intricate connections between Ramanujan’s theta functions and a class of partition functions, Banerjee and Dastidar [‘Ramanujan’s theta functions and parity of parts and cranks of partitions’, Ann. Comb., to appear] studied some arithmetic properties for $c_o(n)$, the number of partitions of n with odd crank. They conjectured a congruence modulo $4$ satisfied by $c_o(n)$. We confirm the conjecture and evaluate $c_o(4n)$ modulo $8$ by dissecting some q-series into even powers. Moreover, we give a conjecture on the density of divisibility of odd cranks modulo 4, 8 and 16.

A partition of a positive integer n is called $\ell $-regular if none of its parts is divisible by $\ell $. Denote by $b_{\ell }(n)$ the number of $\ell $-regular partitions of n. We give a complete characterisation of the arithmetic of $b_{23}(n)$ modulo $11$ for all n not divisible by $11$ in terms of binary quadratic forms. Our result is obtained by establishing a relation between the generating function for these values of $b_{23}(n)$ and certain modular forms having complex multiplication by ${\mathbb Q}(\sqrt {-69})$.

Andrews [Generalized Frobenius Partitions, Memoirs of the American Mathematical Society, 301 (American Mathematical Society, Providence, RI, 1984)] defined two families of functions, $\phi _k(n)$ and $c\phi _k(n),$ enumerating two types of combinatorial objects which he called generalised Frobenius partitions. Andrews proved a number of Ramanujan-like congruences satisfied by specific functions within these two families. Numerous other authors proved similar results for these functions, often with a view towards a specific choice of the parameter $k.$ Our goal is to identify an infinite family of values of k such that $\phi _k(n)$ is even for all n in a specific arithmetic progression; in particular, we prove that, for all positive integers $\ell ,$ all primes $p\geq 5$ and all values $r, 0 < r < p,$ such that $24r+1$ is a quadratic nonresidue modulo $p,$

$$ \begin{align*} \phi_{p\ell-1}(pn+r) \equiv 0 \pmod{2} \end{align*} $$

for all $n\geq 0.$ Our proof of this result is truly elementary, relying on a lemma from Andrews’ memoir, classical q-series results and elementary generating function manipulations. Such a result, which holds for infinitely many values of $k,$ is rare in the study of arithmetic properties satisfied by generalised Frobenius partitions, primarily because of the unwieldy nature of the generating functions in question.

We prove a divisibility result on sums involving the Apéry-like polynomials

$$ \begin{align*} V_n(x)=\sum_{k=0}^n {n\choose k}{n+k\choose k}{x\choose k}{x+k\choose k}, \end{align*} $$

which confirms a conjectural congruence of Z.-H. Sun. Our proof relies on some combinatorial identities and transformation formulae.

The notion of the truncated Euler characteristic for Iwasawa modules is a generalization of the the usual Euler characteristic to the case when the Selmer groups are not finite. Let p be an odd prime, $E_{1}$ and $E_{2}$ be elliptic curves over a number field F with semistable reduction at all primes $v|p$ such that the $\operatorname {Gal}(\overline {F}/F)$ -modules $E_{1}[p]$ and $E_{2}[p]$ are irreducible and isomorphic. We compare the Iwasawa invariants of certain imprimitive multisigned Selmer groups of $E_{1}$ and $E_{2}$ . Leveraging these results, congruence relations for the truncated Euler characteristics associated to these Selmer groups over certain $\mathbb {Z}_{p}^{m}$ -extensions of F are studied. Our results extend earlier congruence relations for elliptic curves over $\mathbb {Q}$ with good ordinary reduction at p.

Let r be an integer with 2 ≤ r ≤ 24 and let pr(n) be defined by $\sum _{n=0}^\infty p_r(n) q^n = \prod _{k=1}^\infty (1-q^k)^r$. In this paper, we provide uniform methods for discovering infinite families of congruences and strange congruences for pr(n) by using some identities on pr(n) due to Newman. As applications, we establish many infinite families of congruences and strange congruences for certain partition functions, such as Andrews's smallest parts function, the coefficients of Ramanujan's ϕ function and p-regular partition functions. For example, we prove that for n ≥ 0,

\[ \textrm{spt}\bigg( \frac{1991n(3n+1) }{2} +83\bigg) \equiv \textrm{spt}\bigg(\frac{1991n(3n+5)}{2} +2074\bigg) \equiv 0\ (\textrm{mod} \ 11), \]

\[ \textrm{spt}\bigg( \frac{143\times 5^{6k} +1 }{24}\bigg)\equiv 2^{k+2} \ (\textrm{mod}\ 11), \]

Let $n$ be either $2$ or an odd integer greater than $1$, and fix a prime $p>2(n+1)$. Under standard ‘adequate image’ assumptions, we show that the set of components of $n$-dimensional $p$-adic potentially semistable local Galois deformation rings that are seen by potentially automorphic compatible systems of polarizable Galois representations over some CM field is independent of the particular global situation. We also (under the same assumption on $n$) improve on the main potential automorphy result of Barnet-Lamb et al. [Potential automorphy and change of weight, Ann. of Math. (2) 179(2) (2014), 501–609], replacing ‘potentially diagonalizable’ by ‘potentially globally realizable’.

For positive integers $n$ and $k$, let $r_{k}(n)$ denote the number of representations of $n$ as a sum of $k$ squares, where representations with different orders and different signs are counted as distinct. For a given positive integer $m$, by means of some properties of binomial coefficients, we derive some infinite families of congruences for $r_{k}(n)$ modulo $2^{m}$. Furthermore, in view of these arithmetic properties of $r_{k}(n)$, we establish many infinite families of congruences for the overpartition function and the overpartition pair function.

Recently, Brietzke, Silva and Sellers [‘Congruences related to an eighth order mock theta function of Gordon and McIntosh’, J. Math. Anal. Appl.479 (2019), 62–89] studied the number $v_{0}(n)$ of overpartitions of $n$ into odd parts without gaps between the nonoverlined parts, whose generating function is related to the mock theta function $V_{0}(q)$ of order 8. In this paper we first present a short proof of the 3-dissection for the generating function of $v_{0}(2n)$. Then we establish three congruences for $v_{0}(n)$ along certain progressions which are subsequences of the integers $4n+3$.

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 two conjectural congruences on the $(p-1)$th Apéry number, which were recently proposed by Z.-H. Sun.

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.