A congruence between the divisor sums σ₃ and σ₅

For every n, the integer 5 σ₃(n) + 7 σ₅(n) is divisible by 12. Termwise, 5 d³ + 7 d⁵ = d³ (5 + 7 d²) is divisible by 12 for every d: modulo 3 either d ≡ 0 or d² ≡ 1, and modulo 4 either d is even, so 8 ∣ d³, or d² ≡ 1; in both cases 5 + 7 = 12.

This is what makes -(5 s₃(q) + 7 s₅(q)) / 12, the coefficient a₆ of the Tate curve, a power series with integer coefficients, so that the Tate curve is defined over ℤ⟦q⟧ and specialises to every ring, residue characteristics 2 and 3 included.

Main results

• TauCeti.twelve_dvd_five_mul_sigma_three_add_seven_mul_sigma_five: 12 ∣ 5 σ₃(n) + 7 σ₅(n).

References

• J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, GTM 151, §V.3.

theorem TauCeti.twelve_dvd_five_mul_sigma_three_add_seven_mul_sigma_five (n : ℕ) : 12 ∣ 5 * (ArithmeticFunction.sigma 3) n + 7 * (ArithmeticFunction.sigma 5) n

5 σ₃(n) + 7 σ₅(n) is divisible by 12.