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奇数の完全数は存在するか？

Does an odd perfect number exist?

mathematicsnumber-theoryperfect-numbers

問題文

自分自身を除く約数の和が自分自身に等しい奇数が存在するかを決定せよ。

Decide whether there is an odd positive integer equal to the sum of its proper divisors.

背景

Open since antiquity. Even perfect numbers are completely characterised by the Euclid-Euler theorem; for odd ones, the literature consists almost entirely of lower bounds and structural constraints that shrink the search space without closing it.

アプローチ · 3 件

解決したかどうかは、問いではなくアプローチごとに決まります。Atlas は判定しません。外部の判定者が何をしたかを記録します。

  • Lean 4 formalization (formal-conjectures)

    形式証明

    到達状況
    未着手
    判定者
    the Lean 4 typechecker, against the pinned toolchain · 型検査
    定式化
    Every perfect number is even, stated in Lean 4 (the negation of the existence question).問いと同値Lean v4.33.1 + mathlib@0df444a3 (formal-conjectures@fc2696b2)検証対象を見る

    保たれているもの

    • The arithmetic content

    加えられた仮定・条件

    • Stated as 'every perfect number is even' rather than as an existence question, which is the same statement contrapositively

    Statement only.

  • Constraints and lower bounds

    査読付きの証明

    到達状況
    進行中
    判定者
    the number theory community, through journal peer review · 査読
    定式化
    Accumulate necessary conditions (size, number of distinct prime factors, special form) until they become contradictory.問いより弱い

    保たれているもの

    • Every constraint holds of any odd perfect number that exists

    弱まっているもの

    • Constraints shrink the search space; they have never been shown to be jointly unsatisfiable

    加えられた仮定・条件

    • Nothing

    Known: any odd perfect number exceeds 10^1500 and has at least 10 distinct prime factors.

    閉じた道

    • Lower bounds and factor counts cannot by themselves prove nonexistence: each new constraint removes a finite region of an infinite search space.無条件

    証拠

  • Exhaustive search

    計算による判定

    到達状況
    進行中
    判定者
    execution against published factorisation campaigns · 実行
    定式化
    Rule out odd perfect numbers below an explicit bound.問いより弱い

    保たれているもの

    • Definitive for the range covered

    弱まっているもの

    • The range is finite and the question is not

    加えられた仮定・条件

    • Trust in the search and the factorisation data

    閉じた道

    • A finite search can only refute the existence question by finding an example; it cannot prove nonexistence.無条件

    証拠

出典

記録

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https://atlasalt.com/q/c4508bbb-4dd6-4cfb-b4c6-821f77c330c0
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2026-09-17
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2026-09-19
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