arXiv subjects

Hajrudin Fejzić

Publications and source records attributed to Hajrudin Fejzić.

4 recordsLinked to original sources

arXiv2026

Nontrivial Solutions to a Cubic Identity and the Factorization of $n^2+n+1$

We investigate a variation of Nicomachus's identity in which one term in the cubic sum is replaced by a different cube. Specifically, we study the Diophantine identity \[ \sum_{j=1}^{n} j^3 + x^3 - k^3 = \left( \sum_{j=1}^{n} j + x - k \right)^2 \] and classify all integer solutions $(k,x,n)$. A full parametric family of nontrivial solutions was introduced in a 2005 paper, along with a conjectural condition for when such solutions exist. We provide a complete proof of this characterization and show it is equivalent to a structural condition on the prime factorization of $ n^2 + n + 1 $. Our argument connects this identity to classical results in the theory of binary quadratic forms. In particular, we analyze the equation $a^2 + ab + b^2 = n^2 + n + 1$, interpreting it as a norm in the ring of Eisenstein integers $\mathbb{Z}[ω]$, where $ω= \frac{1 + \sqrt{-3}}{2}$. This yields a surprising connection between a modified combinatorial identity and the arithmetic of algebraic number fields.

math.GM↗

arXiv2025

On Wegner's 8-Coloring Theorem for Planar Graphs of Maximum Degree Three

We provide a simplified proof of the following special case of Wegner's conjecture: every planar graph of maximum degree at most three admits a distance-2 coloring with at most eight colors. Our main contribution is significant simplification of the most technically challenging part of Wegner's proof: the case involving the removal of a 5-cycle.

math.CO↗

arXiv2025

The Marcinkiewicz-Zygmund Property for Riemann Differences with Geometric Nodes

We study when a Riemann difference of order $ n $ possesses the Marcinkiewicz-Zygmund (MZ) property: that is, whether the conditions $ f(h) = o(h^{n-1}) $ and $ Df(h) = o(h^n) $ imply $ f(h) = o(h^n) $. This implication is known to hold for some classical examples with geometric nodes, such as $ \{0, 1, q, \dots, q^{n-1}\} $ and $ \{1, q, \dots, q^n\} $, leading to a conjecture that these are the only such Riemann differences with the MZ property. However, this conjecture was disproved by the third-order example with nodes $ \{-1, 0, 1, 2\} $, and we provide further counterexamples and a general classification here. We establish a complete analytic criterion for the MZ property by developing a recurrence framework: we analyze when a function $ R(h) $ satisfying $ D(h) = R(qh) - A R(h) $, together with $ D(h) = o(h^n) $ and $ R(h) = o(h^{n-1}) $, forces $ R(h) = o(h^n) $. We prove that this holds if and only if $ A $ lies outside a critical modulus annulus determined by $ q $ and $ n $, covering both $ |q| > 1 $ and $ |q| < 1 $ cases. This leads to a complete characterization of all Riemann differences with geometric nodes that possess the MZ property, and provides a flexible analytic framework applicable to broader classes of generalized differences.

math.CA↗

arXiv2024

Two Pointwise Characterizations of the Peano Derivative

We provide the first two examples of sets of generalized Riemann derivatives of orders up to $n$, $n\geq 2$, whose simultaneous existence for all functions~$f$ at~$x$ is equivalent to the existence of the $n$-th Peano derivative $f_{(n)}(x)$. In this way, we begin to understand how the theory of Peano derivatives can be explained exclusively in terms of generalized Riemann derivatives, a bold new principle in generalized differentiation. In 1936, J. Marcinkiewicz and A. Zygmund showed that the existence of $f_{(n)}(x)$ is equivalent to the existence of both $f_{(n-1)}(x)$ and the $n$th generalized Riemann derivative $\widetilde{D}_nf(x)$, based at $x,x+h,x+2h,x+2^2h,\ldots ,x+2^{n-1}h$. Our first characterization of $f_{(n)}(x)$ is that its existence is equivalent to the simultaneous existence of $\widetilde{D}_1f(x),\ldots,\widetilde{D}_nf(x)$. Our second characterization is that the existence of $f_{(n)}(x)$ is equivalent to the existence of $\widetilde{D}_1f(x)$ and of all $n(n-1)/2$ forward shifts, \[ D_{k,j}f(x)=\lim_{h\rightarrow 0} h^{-k}\sum_{i=0}^k(-1)^i\binom ki f(x+(k+j-i)h), \] for $j=0,1,\ldots,k-2$, of the $k$-th Riemann derivatives $D_{k,0}f(x)$, for $k=2,\ldots ,n$. The proof of the second result involves an interesting combinatorial algorithm that starts with consecutive forward shifts of an arithmetic progression and yields a geometric progression, using two set-operations: dilation and combinatorial Gaussian elimination. This result proves a variant of a 1998 conjecture by Ginchev, Guerragio and Rocca, predicting the same outcome for backward shifts instead of forward shifts. The conjecture has been recently settled in [5], with a proof that has this variant's proof as a prerequisite.

math.CA↗