On sums of powers of the positive integers (Q2855858)

In the usual fashion denote by \(B_k\) the \(k\)th Bernoulli number, by \(B_k(x)\) the \(k\)th Bernoulli polynomial, and let \(S_k(n)=1^k+2^k+\ldots+(n-1)^k\), so that NEWLINE\[NEWLINE S_k(n)=1^k+2^k+\ldots+(n-1)^k=\frac{1}{k+1}\left (B_{k+1}(n)-B_{k+1}\right ), NEWLINE\]NEWLINE as established by Bernoulli for the sums of \(k\)th powers of positive integers, originally considered by Faulhaber.NEWLINENEWLINEIn this paper the author addresses the intriguing request of W. Bednarek, which asks for a characterisation of pairs of positive integers \((k,m)\), such that for every positive integer \(n\), the \textit{divisibility property} NEWLINE\[NEWLINE 1^k+2^k+\ldots+n^k \mid 1^{km}+2^{km}+\ldots + n^{km}, NEWLINE\]NEWLINE is satisfied.NEWLINENEWLINEThe main result states (Theorem 1) that if the \textit{divisibility property} holds for every positive integer \(n\), then \(m\) is odd, and NEWLINE\[NEWLINE B_{km}/B_k\in \mathbb{Z}\,\, \text{for \(k\) even},\qquad m B_{km-1}/B_{k-1}\in \mathbb{Z}\,\, \text{for \(k\) odd \(\geq3\),} NEWLINE\]NEWLINE where the condition is sufficient for \(k\leq 3\), but insufficient for \(k=4\) and infinitely many \(m\).NEWLINENEWLINEThe author then proposes the conjecture that for \(k>3\) the \textit{divisibility property} holds for every positive integer \(n\) only for \(m=1\). This conjecture is supported by the notable counter-example (Theorem 2), that for \(k=4\), and \(n\equiv 58966743\pmod{5^6.11251^2}\), the divisibility property holds only for \(m=1\), and the result (Theorem 3) that for \(m=n=3\), the \textit{divisibility property} holds only for \(k\leq 3\).NEWLINENEWLINEFurther results used to derive the proofs to the above theorems include Lemma 2, which states that if \(P,Q\in\mathbb{Q}[x]\) and \(P(n)/Q(n)\in\mathbb{Z}\) for all sufficiently large integers \(n\), then \(r(x)=P(x)/Q(x)\) is an integer valued polynomial; Lemma 3, which states that if \(3^\nu\|2N\), where \(N=n, n+1\) or \(n+1/2\) and \(\nu\geq 1\), then for every positive integer \(m\), NEWLINE\[NEWLINE 3^{\nu-1}\mid S_{2m}(n+1), NEWLINE\]NEWLINE and Lemma 4, which states that if \(2^\nu\|N\), where \(N=n\) or \(n+1\) and \(\nu\geq 1\), then for every positive integer \(r>2\). NEWLINE\[NEWLINE 2^{\nu-1}\mid S_{2r}(n+1), NEWLINE\]NEWLINE where it is noted that the lemma also holds for \(r\leq2\).NEWLINENEWLINEA minor typo appears in the final display (right) of page 214 where the subscript should have a factor \(m\) included to read \(S_{km+1}(n)\).