Dandapat–Hunsucker–Pomerance (1975): $\sigma(p^k)=2m^2$ and $\sigma(m^2)=p^k$ are incompatible

Proved

OddPerfectNumber.dandapat_hunsucker_pomerance_two

by Gabewhigham · Sep 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

divisor-sumsnumber-theoryperfect-numbers

Write $\sigma(n)=\sum_{d\mid n} d$ for the sum-of-divisors function.

Let $p$ be a prime, let $k \ge 1$ and let $m$ be odd. Then the pair of equations

\sigma(p^{k}) = 2m^{2}, \qquad \sigma(m^{2}) = p^{k}

has no simultaneous solution.

Context. If $N = p^{k}m^{2}$ is an odd perfect number in Euler form, the Dris parametrisation writes $2m^{2} = \sigma(p^{k})s$ and $\sigma(m^{2}) = p^{k}s$ for a positive integer $s$ , the index of the solution. The value $s = 1$ is the extremal case $m^{2} \le p^{k}$ , and it is exactly the pair of equations displayed above. Suryanarayana asked whether an odd perfect number must satisfy them; the statement here says it cannot, for any exponent $k$ , so that every hypothetical odd perfect number has index $s \ge 2$ .

This is the case $t = 2$ , $n = m^{2}$ of Theorem 1 of Dandapat, Hunsucker and Pomerance (1975), which determines all solutions of $\sigma(n) = p^{a}$ , $\sigma(p^{a}) = tn$ : they are $(n,p,a,t) = (21,2,5,3)$ and $n = 2^{c}$ , $p = 2^{c+1}-1$ , $a = 1$ , $t = 2$ . Neither is of the above shape with $m$ odd.

Preamble

import Mathlib
open Finset

Formal statement

namespace OddPerfectNumber

theorem dandapat_hunsucker_pomerance_two (p k m : ℕ) (hp : p.Prime) (hm : Odd m) (hk : k ≠ 0)
    (h1 : (∑ d ∈ (p ^ k).divisors, d) = 2 * m ^ 2)
    (h2 : (∑ d ∈ (m ^ 2).divisors, d) = p ^ k) : False := by sorry

end OddPerfectNumber

Source

G. G. Dandapat, J. L. Hunsucker and C. Pomerance, Some new results on odd perfect numbers, Pacific J. Math. 57 (1975), 359-364, Theorem 1 and its Corollary (case t = 2, n = m^2; the equations (1) of the introduction, raised as a question by Suryanarayana).

View graph

Dandapat–Hunsucker–Pomerance (1975): σ(pk)=2m2\sigma(p^k)=2m^2σ(pk)=2m2 and σ(m2)=pk\sigma(m^2)=p^kσ(m2)=pk are incompatible

Dandapat–Hunsucker–Pomerance (1975): $\sigma(p^k)=2m^2$ and $\sigma(m^2)=p^k$ are incompatible