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A former participant of math olypmiads. I started to think about mathematics lately.

Most recent interests:

  • Gauge integrals: did you know that there exists an integral, whose definition is a generalization of Riemann's definition, yet its strength exceeds that of Lebesgue integral? Instead of limiting your partitions by a positive constant $\delta$: $t_{k+1}-t_k <\delta$, you allow restrictions by a positive function (the "guage") $\delta$: $t_{k+1} - t_{k} < \delta(c_k)$, where $c_k \in [t_k,t_{k+1}]$.
  • Primitive parts of sequences: Take a integer linear recurrence sequence $a_n$ arising from a monic quadratic equation in $\mathbb{Z}[x]$: $x^2 + ax +b = 0$. Examples: Fibonacci, Lucas, $\frac{a^n-b^n}{a-b}$ (where either both $a,b$ are integers, or are conjugate algebraic integers). Then: 1) Under some conditions, $a_n$ has a primitive divisor - a prime divisiors not dividing the previous elements (a statement which translates into statement about order of elements) - that's Zsigmondy's Theorem. 2) $\prod_{d|n} a_d^{\mu(\frac{n}{d})}$ is an integer sequence! For $\frac{a^n-b^n}{a-b}$ it yields the Cyclotomic homogeneous polynomials, and for Fibonacci it yields a sequence called its primitive part.

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