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Question 21701: Complete Prime Multiplicities Determine the Mobius Value

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0042-21701 · q-20261014-advance-20261109-r06-mobius-function-0001

Original question and givens

Use positive integer n=2. Give (n, prime_multiset, distinct_primes, mu, certificate).

  • G001 — Use positive integer n=2. Give (n, prime_multiset, distinct_primes, mu, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source operation=mobius-canonical-value;original snapshot and mathematics retained

After

explicit ordered source output fields=[n, prime_multiset, distinct_primes, mu, certificate]

Conditions: none

Keep the original historical interface marker while listing ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their source claim scope.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

mobius original n:actual positive input for mu=2

After

complete prime multiplicities required for this input;mu(1) uses empty product,not a partial factor search

Conditions: none

Identify whether the input is the original n or one of its positive divisors.Every mu evaluation requires complete prime multiplicities,not a partial factor search.

Step identifiers and dependencies

Rule: mobius_complete_UPF_input_role · Uses: G001, S001

S003 · implies

Before

ORIGINAL positive integer absolute magnitude=2>1

After

initialize remaining=n,prime multiset empty;invariant product(extracted)*remaining=n;every smaller trial divisor has been exhausted

Conditions: none

Divide only by an actual divisor and exhaust smaller candidates,which justifies primality of each extracted divisor without assuming n is prime.

Step identifiers and dependencies

Rule: upf_factorization_remaining_product_invariant · Uses: G001, S002

S004 · calculation

Before

upf canonical complete product factor0 := mul(1, 2)

After

upf canonical complete product factor0 = 2

Conditions: none

Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

exhausted trial divisors below2;terminal remaining=2;d*d>remaining;ORIGINAL n=2

After

complete increasing prime multiset=[2];full product=2;a composite terminal remainder would have an already tested smaller divisor

Conditions: none

If a terminal remainder greater than1 were composite,its least prime factor would already have been exhausted,a contradiction.The final entry is therefore prime.This certifies this instance;general existence follows by strong induction,recursively splitting a composite into smaller positive factors.

Step identifiers and dependencies

Rule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S004

S006 · implies

Before

mobius original n:input=2;COMPLETE prime multiset=[2];distinct primes=[2]

After

mu=-1;empty product case=No;repeated prime=No

Conditions: none

The empty prime multiset for1 gives mu1=1.A repeated prime gives a square divisor and value0;otherwise distinct-prime parity determines the sign.Sorting must not discard multiplicities.

Step identifiers and dependencies

Rule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S005

Reverse checks

  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(2, [2], [2], -1, squarefree_parity_sign)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: a42dc3a7a33f6659fc724ca6839b64724541b8df0f99402f53e4a71f4bd90187
  • candidate record SHA-256: 7d67388186f62eebdaf10cb835f9d2f98dca1a14ff49f57b719e2fef34b51023
  • semantic review receipt SHA-256: dd44555041dfbb6fe9936aa00b9a5f8c5982c2784ef61dd0cbd5d1a303a49c1a
  • question-first witness SHA-256: 9a6011263f6ac53b9a1fd57addf2f3e2f6088b54802c539f332d689480dc2d17
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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