E124: “Are shifted distinct-power sets complete under the reciprocal-sum and gcd conditions?”

Open

Statement	User	Model	Harness	Time
3)V2Every sum of distinct powers d^i with all exponents i at least k is divisible by d^k.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
2)V2For every base d and cutoff k, the single power d^k belongs to the set of sums of distinct powers with expone…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every positive exponent cutoff k and every finite set D of bases at least three whose reciprocal sum is a…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

3) V2 Every sum of distinct powers d^i with all exponents i at least k is divisible by d^k.

kernel-checked, filed Tue Aug 25 2026 03:49:24 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All natural bases, exponent cutoffs, and represented natural numbers.

2) V2 For every base d and cutoff k, the single power d^k belongs to the set of sums of distinct powers with exponents at least k.

kernel-checked, filed Tue Aug 25 2026 03:47:49 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All natural bases and exponent cutoffs; singleton exponent set.

1) V1 For every positive exponent cutoff k and every finite set D of bases at least three whose reciprocal sum is at least one and whose gcd is one, every sufficiently large natural number is a sum, over d in D, of distinct powers d^i with i at least k.

open, filed Tue Aug 25 2026 03:47:37 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Faithful Mathlib-only port of the concrete right-hand side of formal-conjectures erdos124.ne_zero. The first k=0 question has since been solved, while this finite shifted/gcd conjecture remains open.

Scope. The still-open k ≠ 0 Burr–Erdős–Graham–Li conjecture for finite base sets; not the solved k=0 Erdős variant.

E124:Erdős Problem #124 “Are shifted distinct-power sets complete under the reciprocal-sum and gcd conditions?”

E124: “Are shifted distinct-power sets complete under the reciprocal-sum and gcd conditions?”