2^a 3^b = 1 forces a = b = 0

Proposition 4 of the su7-certified-gap paper: with integer exponents, 2^a 3^b = 1 only for a = b = 0; hence ΔU log 2 + ΔV log 3 = 0 has only the trivial integer solution and the fibre of 81 is exact (§5). The proof reads the 2-adic and the 3-adic valuations of both sides.

Main results

• two_zpow_mul_three_zpow_eq_one: (2 : ℚ)^a * 3^b = 1 → a = 0 ∧ b = 0 for a b : ℤ.

References

• C. Marín, A certified gap for SU(7) grand gauge-Higgs unification, Proposition 4 (prop:four).

theorem GaugeHiggsUnification.TwoThreeIndependence.two_zpow_mul_three_zpow_eq_one {a b : ℤ} (h : 2 ^ a * 3 ^ b = 1) : a = 0 ∧ b = 0

2^a 3^b = 1 with integer exponents forces a = b = 0 (compare the 2-adic and 3-adic valuations).