All sufficiently large integers can be written as $a + b + c$ where $a$ has only the digits $0, 1$
in base $3$, $b$ only the digits $0, 1$ in base $4$, $c$ only the digits $0, 1$ in base $7$.
Provee by Burr, Erdős, Graham, and Li [BEGL96]
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lemma erdos124.ne_zero_three_four_seven {k : ℕ} (hk : k ≠ 0) :
∀ᶠ n in atTop,
n ∈ sumsOfDistinctPowers 3 k + sumsOfDistinctPowers 4 k + sumsOfDistinctPowers 7 k
