# Question 9164: Every Prime in an Inclusive Bounded Interval

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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.

R0017-09164 · q-20261013-advance-20261018-r04-prime-number-0089



## Original question and givens

Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=65, hi=85, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).


- G001 — Use natural-number primality: 0 and 1 are not prime. Trial division must cover all candidates through the square-root bound; an incomplete sample is not a primality certificate. lo=65, hi=85, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).





## 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-prime-number](https://ocme.evemisslab.com/en/mko/mko-prime-number/)





## Step-by-step derivation

S001 · impliesBefore

natural-number prime protocol;0 and1 are not prime

After

a failed candidate witness alone is not a primality proof

Conditions: none

Naturals include0.Primality retains both n≥2 and a complete factor bound.A failed single factor candidate only invalidates that witness;it does not certify primality.

Step identifiers and dependenciesRule: prime_complete_evidence_scope · Uses: G001





S002 · impliesBefore

phase=interval value65;n=65

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S001





S003 · impliesBefore

interval value65 trial divisor2:natural dividend=65;positive divisor=2

After

65=32·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S002





S004 · impliesBefore

interval value65 trial divisor3:natural dividend=65;positive divisor=3

After

65=21·3+2

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S003





S005 · impliesBefore

interval value65 trial divisor4:natural dividend=65;positive divisor=4

After

65=16·4+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S004





S006 · impliesBefore

interval value65 trial divisor5:natural dividend=65;positive divisor=5

After

65=13·5+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S005





S007 · impliesBefore

phase=interval value65;tried ascending divisors=[2, 3, 4, 5];sqrt floor=8

After

least factor at least2=5;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S006





S008 · impliesBefore

phase=interval value66;n=66

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S007





S009 · impliesBefore

interval value66 trial divisor2:natural dividend=66;positive divisor=2

After

66=33·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S008





S010 · impliesBefore

phase=interval value66;tried ascending divisors=[2];sqrt floor=8

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S009





S011 · impliesBefore

phase=interval value67;n=67

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S010





S012 · impliesBefore

interval value67 trial divisor2:natural dividend=67;positive divisor=2

After

67=33·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S011





S013 · impliesBefore

interval value67 trial divisor3:natural dividend=67;positive divisor=3

After

67=22·3+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S012





S014 · impliesBefore

interval value67 trial divisor4:natural dividend=67;positive divisor=4

After

67=16·4+3

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S013





S015 · impliesBefore

interval value67 trial divisor5:natural dividend=67;positive divisor=5

After

67=13·5+2

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S014





S016 · impliesBefore

interval value67 trial divisor6:natural dividend=67;positive divisor=6

After

67=11·6+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S015





S017 · impliesBefore

interval value67 trial divisor7:natural dividend=67;positive divisor=7

After

67=9·7+4

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S016





S018 · impliesBefore

interval value67 trial divisor8:natural dividend=67;positive divisor=8

After

67=8·8+3

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S017





S019 · impliesBefore

phase=interval value67;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8];sqrt floor=8

After

least factor at least2=67;prime=Yes

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S018





S020 · impliesBefore

phase=interval value68;n=68

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S019





S021 · impliesBefore

interval value68 trial divisor2:natural dividend=68;positive divisor=2

After

68=34·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S020





S022 · impliesBefore

phase=interval value68;tried ascending divisors=[2];sqrt floor=8

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S021





S023 · impliesBefore

phase=interval value69;n=69

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S022





S024 · impliesBefore

interval value69 trial divisor2:natural dividend=69;positive divisor=2

After

69=34·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S023





S025 · impliesBefore

interval value69 trial divisor3:natural dividend=69;positive divisor=3

After

69=23·3+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S024





S026 · impliesBefore

phase=interval value69;tried ascending divisors=[2, 3];sqrt floor=8

After

least factor at least2=3;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S025





S027 · impliesBefore

phase=interval value70;n=70

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S026





S028 · impliesBefore

interval value70 trial divisor2:natural dividend=70;positive divisor=2

After

70=35·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S027





S029 · impliesBefore

phase=interval value70;tried ascending divisors=[2];sqrt floor=8

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S028





S030 · impliesBefore

phase=interval value71;n=71

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S029





S031 · impliesBefore

interval value71 trial divisor2:natural dividend=71;positive divisor=2

After

71=35·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S030





S032 · impliesBefore

interval value71 trial divisor3:natural dividend=71;positive divisor=3

After

71=23·3+2

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S031





S033 · impliesBefore

interval value71 trial divisor4:natural dividend=71;positive divisor=4

After

71=17·4+3

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S032





S034 · impliesBefore

interval value71 trial divisor5:natural dividend=71;positive divisor=5

After

71=14·5+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S033





S035 · impliesBefore

interval value71 trial divisor6:natural dividend=71;positive divisor=6

After

71=11·6+5

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S034





S036 · impliesBefore

interval value71 trial divisor7:natural dividend=71;positive divisor=7

After

71=10·7+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S035





S037 · impliesBefore

interval value71 trial divisor8:natural dividend=71;positive divisor=8

After

71=8·8+7

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S036





S038 · impliesBefore

phase=interval value71;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8];sqrt floor=8

After

least factor at least2=71;prime=Yes

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S037





S039 · impliesBefore

phase=interval value72;n=72

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S038





S040 · impliesBefore

interval value72 trial divisor2:natural dividend=72;positive divisor=2

After

72=36·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S039





S041 · impliesBefore

phase=interval value72;tried ascending divisors=[2];sqrt floor=8

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S040





S042 · impliesBefore

phase=interval value73;n=73

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S041





S043 · impliesBefore

interval value73 trial divisor2:natural dividend=73;positive divisor=2

After

73=36·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S042





S044 · impliesBefore

interval value73 trial divisor3:natural dividend=73;positive divisor=3

After

73=24·3+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S043





S045 · impliesBefore

interval value73 trial divisor4:natural dividend=73;positive divisor=4

After

73=18·4+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S044





S046 · impliesBefore

interval value73 trial divisor5:natural dividend=73;positive divisor=5

After

73=14·5+3

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S045





S047 · impliesBefore

interval value73 trial divisor6:natural dividend=73;positive divisor=6

After

73=12·6+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S046





S048 · impliesBefore

interval value73 trial divisor7:natural dividend=73;positive divisor=7

After

73=10·7+3

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S047





S049 · impliesBefore

interval value73 trial divisor8:natural dividend=73;positive divisor=8

After

73=9·8+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S048





S050 · impliesBefore

phase=interval value73;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8];sqrt floor=8

After

least factor at least2=73;prime=Yes

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S049





S051 · impliesBefore

phase=interval value74;n=74

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S050





S052 · impliesBefore

interval value74 trial divisor2:natural dividend=74;positive divisor=2

After

74=37·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S051





S053 · impliesBefore

phase=interval value74;tried ascending divisors=[2];sqrt floor=8

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S052





S054 · impliesBefore

phase=interval value75;n=75

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S053





S055 · impliesBefore

interval value75 trial divisor2:natural dividend=75;positive divisor=2

After

75=37·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S054





S056 · impliesBefore

interval value75 trial divisor3:natural dividend=75;positive divisor=3

After

75=25·3+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S055





S057 · impliesBefore

phase=interval value75;tried ascending divisors=[2, 3];sqrt floor=8

After

least factor at least2=3;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S056





S058 · impliesBefore

phase=interval value76;n=76

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S057





S059 · impliesBefore

interval value76 trial divisor2:natural dividend=76;positive divisor=2

After

76=38·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S058





S060 · impliesBefore

phase=interval value76;tried ascending divisors=[2];sqrt floor=8

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S059





S061 · impliesBefore

phase=interval value77;n=77

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S060





S062 · impliesBefore

interval value77 trial divisor2:natural dividend=77;positive divisor=2

After

77=38·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S061





S063 · impliesBefore

interval value77 trial divisor3:natural dividend=77;positive divisor=3

After

77=25·3+2

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S062





S064 · impliesBefore

interval value77 trial divisor4:natural dividend=77;positive divisor=4

After

77=19·4+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S063





S065 · impliesBefore

interval value77 trial divisor5:natural dividend=77;positive divisor=5

After

77=15·5+2

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S064





S066 · impliesBefore

interval value77 trial divisor6:natural dividend=77;positive divisor=6

After

77=12·6+5

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S065





S067 · impliesBefore

interval value77 trial divisor7:natural dividend=77;positive divisor=7

After

77=11·7+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S066





S068 · impliesBefore

phase=interval value77;tried ascending divisors=[2, 3, 4, 5, 6, 7];sqrt floor=8

After

least factor at least2=7;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S067





S069 · impliesBefore

phase=interval value78;n=78

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S068





S070 · impliesBefore

interval value78 trial divisor2:natural dividend=78;positive divisor=2

After

78=39·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S069





S071 · impliesBefore

phase=interval value78;tried ascending divisors=[2];sqrt floor=8

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S070





S072 · impliesBefore

phase=interval value79;n=79

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S071





S073 · impliesBefore

interval value79 trial divisor2:natural dividend=79;positive divisor=2

After

79=39·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S072





S074 · impliesBefore

interval value79 trial divisor3:natural dividend=79;positive divisor=3

After

79=26·3+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S073





S075 · impliesBefore

interval value79 trial divisor4:natural dividend=79;positive divisor=4

After

79=19·4+3

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S074





S076 · impliesBefore

interval value79 trial divisor5:natural dividend=79;positive divisor=5

After

79=15·5+4

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S075





S077 · impliesBefore

interval value79 trial divisor6:natural dividend=79;positive divisor=6

After

79=13·6+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S076





S078 · impliesBefore

interval value79 trial divisor7:natural dividend=79;positive divisor=7

After

79=11·7+2

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S077





S079 · impliesBefore

interval value79 trial divisor8:natural dividend=79;positive divisor=8

After

79=9·8+7

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S078





S080 · impliesBefore

phase=interval value79;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8];sqrt floor=8

After

least factor at least2=79;prime=Yes

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S079





S081 · impliesBefore

phase=interval value80;n=80

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S080





S082 · impliesBefore

interval value80 trial divisor2:natural dividend=80;positive divisor=2

After

80=40·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S081





S083 · impliesBefore

phase=interval value80;tried ascending divisors=[2];sqrt floor=8

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S082





S084 · impliesBefore

phase=interval value81;n=81

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S083





S085 · impliesBefore

interval value81 trial divisor2:natural dividend=81;positive divisor=2

After

81=40·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S084





S086 · impliesBefore

interval value81 trial divisor3:natural dividend=81;positive divisor=3

After

81=27·3+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S085





S087 · impliesBefore

phase=interval value81;tried ascending divisors=[2, 3];sqrt floor=9

After

least factor at least2=3;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S086





S088 · impliesBefore

phase=interval value82;n=82

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S087





S089 · impliesBefore

interval value82 trial divisor2:natural dividend=82;positive divisor=2

After

82=41·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S088





S090 · impliesBefore

phase=interval value82;tried ascending divisors=[2];sqrt floor=9

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S089





S091 · impliesBefore

phase=interval value83;n=83

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S090





S092 · impliesBefore

interval value83 trial divisor2:natural dividend=83;positive divisor=2

After

83=41·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S091





S093 · impliesBefore

interval value83 trial divisor3:natural dividend=83;positive divisor=3

After

83=27·3+2

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S092





S094 · impliesBefore

interval value83 trial divisor4:natural dividend=83;positive divisor=4

After

83=20·4+3

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S093





S095 · impliesBefore

interval value83 trial divisor5:natural dividend=83;positive divisor=5

After

83=16·5+3

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S094





S096 · impliesBefore

interval value83 trial divisor6:natural dividend=83;positive divisor=6

After

83=13·6+5

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S095





S097 · impliesBefore

interval value83 trial divisor7:natural dividend=83;positive divisor=7

After

83=11·7+6

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S096





S098 · impliesBefore

interval value83 trial divisor8:natural dividend=83;positive divisor=8

After

83=10·8+3

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S097





S099 · impliesBefore

interval value83 trial divisor9:natural dividend=83;positive divisor=9

After

83=9·9+2

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S098





S100 · impliesBefore

phase=interval value83;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8, 9];sqrt floor=9

After

least factor at least2=83;prime=Yes

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S099





S101 · impliesBefore

phase=interval value84;n=84

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S100





S102 · impliesBefore

interval value84 trial divisor2:natural dividend=84;positive divisor=2

After

84=42·2+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S101





S103 · impliesBefore

phase=interval value84;tried ascending divisors=[2];sqrt floor=9

After

least factor at least2=2;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S102





S104 · impliesBefore

phase=interval value85;n=85

After

n>=2=Yes

Conditions: none

A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.

Step identifiers and dependenciesRule: prime_lower_bound_guard · Uses: G001, S103





S105 · impliesBefore

interval value85 trial divisor2:natural dividend=85;positive divisor=2

After

85=42·2+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S104





S106 · impliesBefore

interval value85 trial divisor3:natural dividend=85;positive divisor=3

After

85=28·3+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S105





S107 · impliesBefore

interval value85 trial divisor4:natural dividend=85;positive divisor=4

After

85=21·4+1

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S106





S108 · impliesBefore

interval value85 trial divisor5:natural dividend=85;positive divisor=5

After

85=17·5+0

Conditions: none

With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.

Step identifiers and dependenciesRule: positive_euclidean_division · Uses: G001, S107





S109 · impliesBefore

phase=interval value85;tried ascending divisors=[2, 3, 4, 5];sqrt floor=9

After

least factor at least2=5;prime=No

Conditions: none

If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.

Step identifiers and dependenciesRule: complete_small_factor_bound · Uses: G001, S108





S110 · impliesBefore

inclusive interval=[65, 85]

After

all primes=[67, 71, 73, 79, 83];count=5

Conditions: none

Check both interval endpoints and retain exactly those values passing the complete primality test.This finite enumeration makes no distribution claim outside the interval.

Step identifiers and dependenciesRule: complete_inclusive_prime_interval · Uses: G001, S109









## Reverse checks


- S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S109 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S097 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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 — 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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

(65, 85, [67, 71, 73, 79, 83], 5)

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: 0c511135a5d1c27a789a8a723a789b2d478f6009193f1b654a570f158663532a
- candidate record SHA-256: 6409c0c2d16b2c691ba61aa45df814e3436668c885548b9688ebb33a92f923f2
- semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
- question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261018-r04-prime-number-0089.json)