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import FormalConjecturesUtilSigned digit sums in bases $b \ge 5$
The sequence is defined by $$a(n) = \sum_{k=1}^n (-1)^{\operatorname{isprime}(k)} 10^k$$ where the sign is $-1$ if $k$ is prime, and $1$ if $k$ is not prime.
Generalizing the digit pattern observed in $a(n)$, it was conjectured on the OEIS entry that for any base $b \ge 5$ and any choice of signs $\pm 1$, the absolute value of $\sum_{k=1}^n \pm b^k$ only contains digits in ${0, 1, b-2, b-1}$.
References:
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
namespace OeisA243106open FinsetThe sequence $$a(n) = \sum_{k=1}^n (-1)^{\operatorname{isprime}(k)} 10^k$$ where the sign is $-1$ if $k$ is prime, and $1$ if $k$ is not prime.
def a (n : ℕ) : Int :=
(Icc 1 n).sum fun k : ℕ =>
(if Nat.Prime k then (-1 : Int) else 1) * (10 : Int) ^ k@[category test, AMS 11]
lemma a_1 : a 1 = 10 := ⊢ a 1 = 10 All goals completed! 🐙@[category test, AMS 11]
lemma a_2 : a 2 = -90 := ⊢ a 2 = -90 All goals completed! 🐙@[category test, AMS 11]
lemma a_3 : a 3 = -1090 := ⊢ a 3 = -1090 All goals completed! 🐙@[category test, AMS 11]
lemma a_4 : a 4 = 8910 := ⊢ a 4 = 8910 All goals completed! 🐙@[category test, AMS 11]
lemma a_5 : a 5 = -91090 := ⊢ a 5 = -91090 All goals completed! 🐙Conjecture: When expressed in base $b \ge 5$, the absolute value of any partial sum $\sum \pm b^k$ only contains digits belonging to ${0, 1, b-2, b-1}$.
A formal proof has been found with the methods described in arxiv/2605.22763.
@[category research solved, AMS 11, formal_proof using formal_conjectures at
"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/243106.wip.lean#L140"]
theorem digits_sum_restricted (b n : ℕ) (hb : b ≥ 5) :
∀ (σ : ℕ → Int) (hσ : ∀ k ∈ Icc 1 n, σ k = 1 ∨ σ k = -1),
let x : Int := (Icc 1 n).sum fun k ↦ σ k * (b : Int) ^ k;
∀ d ∈ (b.digits x.natAbs), d = 0 ∨ d = 1 ∨ d = b - 2 ∨ d = b - 1 := b:ℕn:ℕhb:b ≥ 5⊢ ∀ (σ : ℕ → ℤ),
(∀ k ∈ Icc 1 n, σ k = 1 ∨ σ k = -1) →
let x := ∑ k ∈ Icc 1 n, σ k * ↑b ^ k;
∀ d ∈ b.digits x.natAbs, d = 0 ∨ d = 1 ∨ d = b - 2 ∨ d = b - 1
All goals completed! 🐙end OeisA243106