# Diophantine problems with sums of divisors

Bujačić, Sanda (2014) Diophantine problems with sums of divisors. Doctoral thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

Ayad and Luca have proved that there does not exist an odd integer $n > 1$ and two positive divisors $d_{1}$,$d_{2}$ of $(n^{2}+1)/2$ such that $d_{1}+d_{2}=n+1$. Dujella and Luca have dealt with a more general issue, where n + 1 was replaced with an arbitrary linear polynomial $\delta n+\varepsilon$, where $\delta > 0$ and $\varepsilon$ are given integers. The reason that $d_{1}$ and $d_{2}$ are congruent to 1 modulo 4 comes from the fact that $(n^{2}+1)/2$ is odd and is a sum of two coprime squares ${((n+1)/2)}^2+{((n-1)/2)}^2$. Such numbers have the property that all their prime factors are congruent to 1 modulo 4. Since $d_{1}+d_{2}=\delta n+ \varepsilon$, then there are two cases: it is either $\delta \equiv \varepsilon \equiv 1 \pmod{2}$, or $\delta \equiv \varepsilon +2 \equiv 0$ or $2 \pmod{4}$. Dujella and Luca have focused on the first case. We deal with the second case, the case where $\delta \equiv \varepsilon +2 \equiv 0$ or $2 \pmod{4}$. We completely solve cases when $\delta =2, \delta =4$ and $\varepsilon =0$. We prove that there exist infinitely many positive odd integers n with the property that there exists a pair of positive divisors $d_{1}$,$d_{2}$ of $(n^{2}+1)/2$ such that $d_{1}+d_{2}=2n+ \varepsilon$ for $\varepsilon \equiv 0 \pmod{4}$ and we prove an analogous result for $\varepsilon \equiv 2 \pmod{4}$ and divisors $d_{1},d_{2}$ of $(n^{2}+1)/2$ such that $d_{1}+d_{2}=4n+ \varepsilon$. In the case when $\delta \geq{6}$ is a positive integer of the form $\delta =4k+2, k \in \mathbb{N}$ we prove that there does not exist an odd integer n such that there exists a pair of divisors $d_{1},d_{2}$ of $(n^{2}+1)/2$ with the property $d_{1}+d_{2}=\delta n$. We also prove that there exist infinitely many odd integers n with the property that there exists a pair of positive divisors $d_{1},d_{2}$ of $(n^{2}+1)/2$ such that $d_{1}+d_{2}=2n$. The second part of the doctoral thesis deals with the similar problem or, more specifically, it deals with one-parametric families of coefficients $\delta$ , $\varepsilon$. We prove that there exist infinitely many odd integers n with the property that there exists a pair of positive divisors $d_{1},d_{2}$ of $(n^{2}+1)/2$ such that $d_{1}+d_{2}=\delta n+(\delta +2)$. We also prove that there exist infinitely many odd integers n with the property that there exists a pair of positive divisors $d_{1},d_{2}$ of $(n^{2}+1)/2$ such that $d_{1}+d_{2}=\delta n+(\delta -2),\delta \equiv{4,6} \pmod{8}$. The third part of the doctoral thesis deals with the version of Subbarao’s congruence, or, more precisely, with the congruence of the form $n\varphi(n) \equiv 2 \pmod{\sigma(n)}$, where $\varphi(n)$ is Euler’s totient function and $\sigma(n)$ is sum of divisors of n. Dujella and Luca have proved that there exist only finitely many integers n whose prime factors belong to a fixed finite set and satisfy the congruence. We prove that the only integers of the form $n=2^{\alpha}5^{\beta}, \alpha, \beta \geq{0}$, that satisfy that congruence are integers n = 1, 2, 5, 8.