# Question 21706: Complete Prime Multiplicities Determine the Mobius Value [Return to catalogue page 109](https://ocme.evemisslab.com/en/questions/page/109/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261109-r06-mobius-function-foundations/#q-20261014-advance-20261109-r06-mobius-function-0006) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0042-21706 · q-20261014-advance-20261109-r06-mobius-function-0006 ## Original question and givens Use positive integer n=42. Give (n, prime_multiset, distinct_primes, mu, certificate). - G001 — Use positive integer n=42. 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. - [mko-mobius-function](https://ocme.evemisslab.com/en/mko/mko-mobius-function/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore mobius original n:actual positive input for mu=42 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 dependenciesRule: mobius_complete_UPF_input_role · Uses: G001, S001 S003 · impliesBefore ORIGINAL positive integer absolute magnitude=42>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 dependenciesRule: upf_factorization_remaining_product_invariant · Uses: G001, S002 S004 · calculationBefore upf extract divisor2 occurrence0 := div(42, 2) After upf extract divisor2 occurrence0 = 21 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore next divisor=2;previous remaining=42;all smaller divisors exhausted After extracted multiplicity=1;next remaining=21;remaining modulo2=1;extracted multiset=[2] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S004 S006 · calculationBefore upf extract divisor3 occurrence1 := div(21, 3) After upf extract divisor3 occurrence1 = 7 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore next divisor=3;previous remaining=21;all smaller divisors exhausted After extracted multiplicity=1;next remaining=7;remaining modulo3=1;extracted multiset=[2, 3] Conditions: none Exhaust every multiplicity before advancing.The stopping test concerns the current remainder,and the final prime remainder must not be omitted. Step identifiers and dependenciesRule: upf_complete_trial_division_remaining_invariant · Uses: G001, S006 S008 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore upf canonical complete product factor1 := mul(2, 3) After upf canonical complete product factor1 = 6 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore upf canonical complete product factor2 := mul(6, 7) After upf canonical complete product factor2 = 42 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore exhausted trial divisors below4;terminal remaining=7;d*d>remaining;ORIGINAL n=42 After complete increasing prime multiset=[2, 3, 7];full product=42;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 dependenciesRule: upf_final_prime_remainder_complete_existence_instance · Uses: G001, S010 S012 · impliesBefore mobius original n:input=42;COMPLETE prime multiset=[2, 3, 7];distinct primes=[2, 3, 7] 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 dependenciesRule: mobius_value_one_repeated_prime_or_squarefree_parity · Uses: G001, S011 ## Reverse checks - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Recompute upf canonical complete product factor2 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute upf canonical complete product factor1 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute upf canonical complete product factor0 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute upf extract divisor3 occurrence1 with exact rational arithmetic, not rounded decimals. - 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 extract divisor2 occurrence0 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 (42, [2, 3, 7], [2, 3, 7], -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: 454e71aa44704715ae2f76cf37b262d890ec8025773214938f4b54915beeffc2 - candidate record SHA-256: 0107b7aeea7dc7187cdd4a202c90a4b9d5c0aa11802a38527e4ac48e95b6bee9 - semantic review receipt SHA-256: dd44555041dfbb6fe9936aa00b9a5f8c5982c2784ef61dd0cbd5d1a303a49c1a - question-first witness SHA-256: 9a6011263f6ac53b9a1fd57addf2f3e2f6088b54802c539f332d689480dc2d17 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261109-r06-mobius-function-0006.json)