26 problems

Eswarathasan–Levine conjecture. There are infinitely many harmonic primes.

Boyd's valuation-four conjecture. The inequality $\nu_p(H_n)\geq 4$ never occurs.

For a positive integer $p$, let $B_r$ denote the Bernoulli numbers, let $H_n=\sum_{k=1}^n 1/k$ be the harmonic numbers, and let $H_n^{(2)}=\sum_{k=1}^n 1/k^2$ be the second-order h…

Let $S_1=S_1(a,b,c,d;n)$ and $S_2=S_2(a,b,c,d;n)$ be the two summands obtained from the WZ seeds in the harmonic extension of the displayed series, and let $V_N(S_i)$ be their degr…

Let $q\geq 1$ be an integer, and let $H_k=\sum_{j=1}^k\frac{1}{j}$ denote the $k$th harmonic number. For rational numbers $\alpha_{q,2},\alpha_{q,3},\ldots,\alpha_{q,q+2}$, conside…

Let $H_n=\frac1{1}+\frac1{2}+\fdots+\frac1n$ denote the $n$th harmonic number, with $H_0=0$, and let $G$ and $K$ denote the constants appearing on the right-hand sides of the ident…

Let $Hn= displaystylesum{i=1}^nfrac{1}{i}$, written in lowest terms as $Hn=un/vn$ with $uninmathbb{Z}$ and $vninmathbb{N}$. Fix a prime $p$. Finite-divisibility conjecture. The…

Let $h_j$ denote the $j$th sequence of iterated harmonic numbers, with Property 5 referring to the corresponding property established earlier in the paper. Property 5 extension con…

For each integer $j\geq 1$, let $h_j(n)$ denote the $j$th iterated harmonic number, as defined in the paper. In particular, $h_j(1)=1$. Integrality conjecture. For each integer…

Sun's three conjectures. The following identities hold:

Let $p(m)$ be a polynomial in $m$ of degree $d$, and let $H_m$ denote the $m$-th harmonic number. For each positive integer $t\geq 4$, suppose there exist polynomials…

Campbell's parbelos-constant conjecture. The hypergeometric value satisfies

Let $H_n=\frac{c_n}{d_n}$ with $\operatorname{GCD}(c_n,d_n)=1$, let $D_n=\operatorname{LCM}(1,2,\ldots,n)$, and define $q_n$ by $D_n=d_nq_n$. Let … where…

Let $H_n=1+\frac12+\cdots+\frac1n=\frac{c_n}{d_n}$ with $\operatorname{GCD}(c_n,d_n)=1$, and let $D_n=\operatorname{LCM}(1,2,\ldots,n)$. Harmonic-denominator conjecture. It is conj…

For integers $n\geq k\geq1$, define … and let $\nu_p$ denote the $p$-adic valuation. The logarithmic lower-bound conjecture. For any prime number $p$ and any integer $k\geq1$, ther…

Let $A_k$ be the Apéry numbers, … For a prime $p>5$, let $H_n=\sum_{0<k\leq n}1/k$ be the $n$-th harmonic number, and let $B_0,B_1,B_2,\ldots$ be the Bernoulli numbers. Sun's conje…

For a positive integer $m$, let $H_k^{(m)}:=\sum_{0<j\leqslant k}1/j^m$ be the $m$-th order harmonic number. The author’s conjecture. For every prime $p\equiv1\pmod4$ and every eve…

Let $H_n=\frac{N_n}{D_n}$ be the $n$th harmonic number written in reduced form, and let $L_n=\operatorname*{lcm}(1,2,\dots,n)$. Graham's conjecture. The equality … holds for infini…

Let $p>3$ be a prime. Write … and let $E_{p-3}$ denote the Euler number of index $p-3$. Then the Euler-number congruence. … The source reports that this conjecture was confirmed by…

Let $p>3$ be a prime, let $H_k=\sum_{j=1}^k\frac1j$ be the $k$th harmonic number, and let $B_{p-5}$ denote the Bernoulli number of index $p-5$. Su's conjecture. … This is the secon…

Let $p$ be a prime, let $N$ be a positive integer, let $H_N=\sum_{n=1}^N\frac1n$ be the $N$-th harmonic number, and let $v_p$ denote the $p$-adic valuation. Shifted harmonic-number…

Let $p$ be a prime, let $N$ be a positive integer, let $H_N=\sum_{n=1}^N\frac1n$ be the $N$-th harmonic number, and let $v_p$ denote the $p$-adic valuation. Harmonic-number valuati…

Let $H_n^{(r)}$ denote the hyperharmonic number of order $r$ and index $n$, with $r,n\ge 2$. Non-integrality conjecture. None of the hyperharmonic numbers can be integers: … The pr…

Let $p$ be a prime and $N$ a positive integer. Let $H_N$ be the $N$-th harmonic number and let $v_p$ denote the $p$-adic valuation. Shifted harmonic-number valuation conjecture. Th…

Let $p$ be a prime and $N$ a positive integer with $p\le N$. Let $H_N$ be the $N$-th harmonic number, let $v_p$ denote the $p$-adic valuation, and call $p$ a Wolstenholme prime whe…