A small D and a sixth power of five force five into the sigma value
ProvedOddPerfectNumber.q2_five_q3_nineteen_q4_127_five_dvd_sigmaarithmetick-oneq2-fiveq3-nineteenq4-127
In the canonical product and sigma equations, if D=(p+1)/2 is below 185 and 5^6 divides m^2, then 5 divides sigma.
Preamble
import Mathlib
Formal statement
namespace OddPerfectNumber
theorem q2_five_q3_nineteen_q4_127_five_dvd_sigma (p m d sigma : Nat)
(hprod : m ^ 2 = ((p + 1) / 2) * d)
(hsigma : sigma = p * d)
(hD : (p + 1) / 2 < 185)
(hpow : 5 ^ 6 ∣ m ^ 2) :
5 ∣ sigma := by
sorry
end OddPerfectNumber
Source
Exact prime-power divisibility and cross-multiplication; no odd-perfect-specific theorem is assumed.