eᵐ is irrational for m∈ℕ

Proof. If eᵐ=a/b with a,b∈ℕ then for any n∈ℕ the integral
b ∫₀¹ {eᵐˣm²ⁿ⁺¹} {xⁿ(1–x)ⁿ/n!} dx
is an integer (use beᵐ=a & integration by parts 2n-times)
But
0 < b ∫₀¹ eᵐˣm²ⁿ⁺¹xⁿ(1–x)ⁿ/n! dx < beᵐm²ⁿ⁺¹/n! < 1
for n large enough.
QED pic.twitter.com/nIorpUtxhX

— Tamás Görbe (@TamasGorbe) April 5, 2019