Let $a(n)$ be the sequence A158690 in the On-Line Encyclopedia of Integer Sequences (OEIS), defined by the exponential generating function $\sum_{n\ge0} a(n)t^n/n! = 1+\sum_{m\ge1}\prod_{j=1}^m(1-e^{-(2j-1)t})$. We prove two congruence conjectures of Peter Bala. The first states that, for every integer $k\ge1$, the sequence $a(n)$ modulo $k$ is eventually periodic with period dividing $\varphi(k)$. We prove the stronger statement that the Carmichael function $\lambda(k)$ is an eventual period. The second conjecture asserts the shifted Gauss congruences $a(np^r+i)\equiv a(np^{r-1}+i)\pmod{p^r}$ for every $i\ge0$, every prime $p$, and all $n,r\ge1$. Both results follow from a general theorem for exponential generating functions of the form $G(e^t-1)$ with $G\in\mathbb Z[[y]]$, together with the standard power-sum formula for Stirling numbers of the second kind.
For integers $r\ge2$ and $n\ge1$, let $$ F_{r,n}(q)=\prod_{k=1}^{n}\frac{1-q^{rk}}{1-q^k}. $$ The coefficient of $q^j$ in \(F_{r,n}(q)\) counts partitions of $j$ into parts at most $n$, each occurring at most $r-1$ times. Hughes proved that $F_{2,n}(q)=\prod_{k=1}^{n}(1+q^k)$ is unimodal for every $n\ge 1$. This result...
Let $(a_n)_{n\geqslant 0}$ be a sequence of integers. Its dual sequence $(a_n^*)_{n\geqslant 0}$ is defined by \begin{equation*} a_n^* := \sum_{k=0}^{n} \binom{n}{k}(-1)^k a_k. \end{equation*} Let $p>3$ be a prime. In this paper we mainly investigate congruences modulo $p^2$ involving central binomial coefficients and...
Let $P_n(q)=\prod_{j=1}^n(1-q^{3j-2})(1-q^{3j-1})$. For each $p\in\{4,5,6,7,8\}$, we prove that the coefficients of $P_n(q)^p$ in residue class $0$ modulo $3$ are nonnegative, and that those in residue class $2$, after zero terms are omitted, change sign exactly once, from positive to negative. The residue-$0$ coeffici...
Let $T(x)=\prod_{j\ge 0}(1-x^{2^j})=\sum_{n\ge 0}t(n)x^n$ be the Thue--Morse generating function, and write $T(x)^m=\sum_{n\ge 0}t_m(n)x^n$ for a positive integer $m$. For a nonzero integer $a$, let $\nu_2(a)$ be the exponent of $2$ in $a$ and put $\operatorname{odd}(a)=a/2^{\nu_2(a)}$. For every $s\ge 1$, we prove tha...
We prove the irreducibility conjecture for Legendre polynomials proposed by Stieltjes: for every integer $j\geq 1$, both $P_{2j}(x)$ and $P_{2j+1}(x)/x$ are irreducible over the rational numbers. The proof proceeds by contradiction. Starting from a hypothetical nontrivial factorization, we remove the zero root from $P_...
The preprint arXiv:2609.04176 (Z.-W. Sun, 3 September 2026) claims a proof that Catalan's constant $G$ is irrational: it constructs for each $B$ a number $\hat q_B$ from weighted tails of the series and asserts that, if $G=a/q$, the integer $N_B=q^S H_B^{\min}\hat q_B$ satisfies $0<|N_B|<1$ for large $B$ (its Theorem 9...
D. Wachs· 0 citations
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