I put forward a hypothesis in number theory, it is as follows.$ \sigma\_1(n)=\sigma\_1(m)=p$, where $\sigma\_1$ is the divisor sum function, $n,m\in \mathbb N$, and $p$ is prime. I recently noticed and assumed that the pattern $ \sigma\_1(16)=\sigma\_1(25)=31$ is the only one. And what do you think about this?



Are there infinite numbers of the form $\sigma_1(n)=\sigma_1(m)=p$, or is there only one?
As mentioned in the comments, it suffices to assume that $m,n$ are prime powers. Then we are looking at an equation of the form


$$\displaystyle u^a + \cdots + u + 1 = v^b + \cdots + v + 1.$$


The expressions on either side are cyclotomic polynomials. If either one is reducible (over $\mathbb{Z}$), then it is exceedingly unlikely that they will evaluate to a prime. Thus we may assume, without losing very much, that both of the cyclotomic polynomials are irreducible. This then requires both $a,b$ to be of the form $p-1$ for some prime $p$.


Note that $16 = 2^4 = 2^{5-1}$ and $25 = 5^2 = 5^{3-1}$ are both of this form.


We are then looking for integer points on a family of plane curves of the form given by


$$\displaystyle x^{p-1} + \cdots + x = y^{q-1} + \cdots + y, x,y \in \mathbb{P}$$


where $\mathbb{P}$ denotes the set of prime numbers.


Results of this type, with one variable exponent, are known in the literature; see for example the papers of Bennett and Dahmen ([Klein forms and the generalized superelliptic equation](https://%20%20%20https://annals.math.princeton.edu/2013/177-1/p04)) and Bennett and Siksek ([Rational points on Erdős–Selfridge superelliptic curves](https://www.cambridge.org/core/journals/compositio-mathematica/article/rational-points-on-erdosselfridge-superelliptic-curves/1C53006D003585C6F4746BE4E0162836)). However, these do not rule out the possible existence of infinitely many solutions with two variable exponents.


It would be interesting to see if the family of hyperelliptic curves corresponding to $q = 3$ in the above family can be treated using the methods in the two papers above. That is, are there only finitely many integers $x,y,p$ (with $p$ variable) such that


$$\displaystyle y^2 + y = x^{p-1} + \cdots + x?$$


