# Question 9175: 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.

R0017-09175 · q-20261013-advance-20261018-r04-prime-number-0100



## 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=120, hi=140, 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=120, hi=140, 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 value120;n=120

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 value120 trial divisor2:natural dividend=120;positive divisor=2

After

120=60·2+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, S002





S004 · impliesBefore

phase=interval value120;tried ascending divisors=[2];sqrt floor=10

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, S003





S005 · impliesBefore

phase=interval value121;n=121

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, S004





S006 · impliesBefore

interval value121 trial divisor2:natural dividend=121;positive divisor=2

After

121=60·2+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, S005





S007 · impliesBefore

interval value121 trial divisor3:natural dividend=121;positive divisor=3

After

121=40·3+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, S006





S008 · impliesBefore

interval value121 trial divisor4:natural dividend=121;positive divisor=4

After

121=30·4+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, S007





S009 · impliesBefore

interval value121 trial divisor5:natural dividend=121;positive divisor=5

After

121=24·5+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, S008





S010 · impliesBefore

interval value121 trial divisor6:natural dividend=121;positive divisor=6

After

121=20·6+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, S009





S011 · impliesBefore

interval value121 trial divisor7:natural dividend=121;positive divisor=7

After

121=17·7+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, S010





S012 · impliesBefore

interval value121 trial divisor8:natural dividend=121;positive divisor=8

After

121=15·8+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, S011





S013 · impliesBefore

interval value121 trial divisor9:natural dividend=121;positive divisor=9

After

121=13·9+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, S012





S014 · impliesBefore

interval value121 trial divisor10:natural dividend=121;positive divisor=10

After

121=12·10+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, S013





S015 · impliesBefore

interval value121 trial divisor11:natural dividend=121;positive divisor=11

After

121=11·11+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, S014





S016 · impliesBefore

phase=interval value121;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8, 9, 10, 11];sqrt floor=11

After

least factor at least2=11;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, S015





S017 · impliesBefore

phase=interval value122;n=122

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, S016





S018 · impliesBefore

interval value122 trial divisor2:natural dividend=122;positive divisor=2

After

122=61·2+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, S017





S019 · impliesBefore

phase=interval value122;tried ascending divisors=[2];sqrt floor=11

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, S018





S020 · impliesBefore

phase=interval value123;n=123

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 value123 trial divisor2:natural dividend=123;positive divisor=2

After

123=61·2+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, S020





S022 · impliesBefore

interval value123 trial divisor3:natural dividend=123;positive divisor=3

After

123=41·3+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, S021





S023 · impliesBefore

phase=interval value123;tried ascending divisors=[2, 3];sqrt floor=11

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, S022





S024 · impliesBefore

phase=interval value124;n=124

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, S023





S025 · impliesBefore

interval value124 trial divisor2:natural dividend=124;positive divisor=2

After

124=62·2+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, S024





S026 · impliesBefore

phase=interval value124;tried ascending divisors=[2];sqrt floor=11

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, S025





S027 · impliesBefore

phase=interval value125;n=125

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 value125 trial divisor2:natural dividend=125;positive divisor=2

After

125=62·2+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, S027





S029 · impliesBefore

interval value125 trial divisor3:natural dividend=125;positive divisor=3

After

125=41·3+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, S028





S030 · impliesBefore

interval value125 trial divisor4:natural dividend=125;positive divisor=4

After

125=31·4+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, S029





S031 · impliesBefore

interval value125 trial divisor5:natural dividend=125;positive divisor=5

After

125=25·5+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, S030





S032 · impliesBefore

phase=interval value125;tried ascending divisors=[2, 3, 4, 5];sqrt floor=11

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, S031





S033 · impliesBefore

phase=interval value126;n=126

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, S032





S034 · impliesBefore

interval value126 trial divisor2:natural dividend=126;positive divisor=2

After

126=63·2+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, S033





S035 · impliesBefore

phase=interval value126;tried ascending divisors=[2];sqrt floor=11

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, S034





S036 · impliesBefore

phase=interval value127;n=127

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, S035





S037 · impliesBefore

interval value127 trial divisor2:natural dividend=127;positive divisor=2

After

127=63·2+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, S036





S038 · impliesBefore

interval value127 trial divisor3:natural dividend=127;positive divisor=3

After

127=42·3+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, S037





S039 · impliesBefore

interval value127 trial divisor4:natural dividend=127;positive divisor=4

After

127=31·4+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, S038





S040 · impliesBefore

interval value127 trial divisor5:natural dividend=127;positive divisor=5

After

127=25·5+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, S039





S041 · impliesBefore

interval value127 trial divisor6:natural dividend=127;positive divisor=6

After

127=21·6+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, S040





S042 · impliesBefore

interval value127 trial divisor7:natural dividend=127;positive divisor=7

After

127=18·7+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, S041





S043 · impliesBefore

interval value127 trial divisor8:natural dividend=127;positive divisor=8

After

127=15·8+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, S042





S044 · impliesBefore

interval value127 trial divisor9:natural dividend=127;positive divisor=9

After

127=14·9+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 value127 trial divisor10:natural dividend=127;positive divisor=10

After

127=12·10+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, S044





S046 · impliesBefore

interval value127 trial divisor11:natural dividend=127;positive divisor=11

After

127=11·11+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, S045





S047 · impliesBefore

phase=interval value127;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8, 9, 10, 11];sqrt floor=11

After

least factor at least2=127;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, S046





S048 · impliesBefore

phase=interval value128;n=128

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, S047





S049 · impliesBefore

interval value128 trial divisor2:natural dividend=128;positive divisor=2

After

128=64·2+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, S048





S050 · impliesBefore

phase=interval value128;tried ascending divisors=[2];sqrt floor=11

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, S049





S051 · impliesBefore

phase=interval value129;n=129

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 value129 trial divisor2:natural dividend=129;positive divisor=2

After

129=64·2+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, S051





S053 · impliesBefore

interval value129 trial divisor3:natural dividend=129;positive divisor=3

After

129=43·3+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, S052





S054 · impliesBefore

phase=interval value129;tried ascending divisors=[2, 3];sqrt floor=11

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, S053





S055 · impliesBefore

phase=interval value130;n=130

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, S054





S056 · impliesBefore

interval value130 trial divisor2:natural dividend=130;positive divisor=2

After

130=65·2+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, S055





S057 · impliesBefore

phase=interval value130;tried ascending divisors=[2];sqrt floor=11

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, S056





S058 · impliesBefore

phase=interval value131;n=131

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 value131 trial divisor2:natural dividend=131;positive divisor=2

After

131=65·2+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, S058





S060 · impliesBefore

interval value131 trial divisor3:natural dividend=131;positive divisor=3

After

131=43·3+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, S059





S061 · impliesBefore

interval value131 trial divisor4:natural dividend=131;positive divisor=4

After

131=32·4+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, S060





S062 · impliesBefore

interval value131 trial divisor5:natural dividend=131;positive divisor=5

After

131=26·5+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, S061





S063 · impliesBefore

interval value131 trial divisor6:natural dividend=131;positive divisor=6

After

131=21·6+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, S062





S064 · impliesBefore

interval value131 trial divisor7:natural dividend=131;positive divisor=7

After

131=18·7+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, S063





S065 · impliesBefore

interval value131 trial divisor8:natural dividend=131;positive divisor=8

After

131=16·8+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, S064





S066 · impliesBefore

interval value131 trial divisor9:natural dividend=131;positive divisor=9

After

131=14·9+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 value131 trial divisor10:natural dividend=131;positive divisor=10

After

131=13·10+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, S066





S068 · impliesBefore

interval value131 trial divisor11:natural dividend=131;positive divisor=11

After

131=11·11+10

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, S067





S069 · impliesBefore

phase=interval value131;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8, 9, 10, 11];sqrt floor=11

After

least factor at least2=131;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, S068





S070 · impliesBefore

phase=interval value132;n=132

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, S069





S071 · impliesBefore

interval value132 trial divisor2:natural dividend=132;positive divisor=2

After

132=66·2+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, S070





S072 · impliesBefore

phase=interval value132;tried ascending divisors=[2];sqrt floor=11

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, S071





S073 · impliesBefore

phase=interval value133;n=133

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, S072





S074 · impliesBefore

interval value133 trial divisor2:natural dividend=133;positive divisor=2

After

133=66·2+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, S073





S075 · impliesBefore

interval value133 trial divisor3:natural dividend=133;positive divisor=3

After

133=44·3+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, S074





S076 · impliesBefore

interval value133 trial divisor4:natural dividend=133;positive divisor=4

After

133=33·4+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, S075





S077 · impliesBefore

interval value133 trial divisor5:natural dividend=133;positive divisor=5

After

133=26·5+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, S076





S078 · impliesBefore

interval value133 trial divisor6:natural dividend=133;positive divisor=6

After

133=22·6+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, S077





S079 · impliesBefore

interval value133 trial divisor7:natural dividend=133;positive divisor=7

After

133=19·7+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, S078





S080 · impliesBefore

phase=interval value133;tried ascending divisors=[2, 3, 4, 5, 6, 7];sqrt floor=11

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, S079





S081 · impliesBefore

phase=interval value134;n=134

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 value134 trial divisor2:natural dividend=134;positive divisor=2

After

134=67·2+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 value134;tried ascending divisors=[2];sqrt floor=11

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 value135;n=135

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 value135 trial divisor2:natural dividend=135;positive divisor=2

After

135=67·2+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 value135 trial divisor3:natural dividend=135;positive divisor=3

After

135=45·3+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 value135;tried ascending divisors=[2, 3];sqrt floor=11

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 value136;n=136

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 value136 trial divisor2:natural dividend=136;positive divisor=2

After

136=68·2+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 value136;tried ascending divisors=[2];sqrt floor=11

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 value137;n=137

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 value137 trial divisor2:natural dividend=137;positive divisor=2

After

137=68·2+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 value137 trial divisor3:natural dividend=137;positive divisor=3

After

137=45·3+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 value137 trial divisor4:natural dividend=137;positive divisor=4

After

137=34·4+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, S093





S095 · impliesBefore

interval value137 trial divisor5:natural dividend=137;positive divisor=5

After

137=27·5+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, S094





S096 · impliesBefore

interval value137 trial divisor6:natural dividend=137;positive divisor=6

After

137=22·6+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 value137 trial divisor7:natural dividend=137;positive divisor=7

After

137=19·7+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, S096





S098 · impliesBefore

interval value137 trial divisor8:natural dividend=137;positive divisor=8

After

137=17·8+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, S097





S099 · impliesBefore

interval value137 trial divisor9:natural dividend=137;positive divisor=9

After

137=15·9+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

interval value137 trial divisor10:natural dividend=137;positive divisor=10

After

137=13·10+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, S099





S101 · impliesBefore

interval value137 trial divisor11:natural dividend=137;positive divisor=11

After

137=12·11+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, S100





S102 · impliesBefore

phase=interval value137;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8, 9, 10, 11];sqrt floor=11

After

least factor at least2=137;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, S101





S103 · impliesBefore

phase=interval value138;n=138

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, S102





S104 · impliesBefore

interval value138 trial divisor2:natural dividend=138;positive divisor=2

After

138=69·2+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, S103





S105 · impliesBefore

phase=interval value138;tried ascending divisors=[2];sqrt floor=11

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, S104





S106 · impliesBefore

phase=interval value139;n=139

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, S105





S107 · impliesBefore

interval value139 trial divisor2:natural dividend=139;positive divisor=2

After

139=69·2+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, S106





S108 · impliesBefore

interval value139 trial divisor3:natural dividend=139;positive divisor=3

After

139=46·3+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, S107





S109 · impliesBefore

interval value139 trial divisor4:natural dividend=139;positive divisor=4

After

139=34·4+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, S108





S110 · impliesBefore

interval value139 trial divisor5:natural dividend=139;positive divisor=5

After

139=27·5+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, S109





S111 · impliesBefore

interval value139 trial divisor6:natural dividend=139;positive divisor=6

After

139=23·6+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, S110





S112 · impliesBefore

interval value139 trial divisor7:natural dividend=139;positive divisor=7

After

139=19·7+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, S111





S113 · impliesBefore

interval value139 trial divisor8:natural dividend=139;positive divisor=8

After

139=17·8+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, S112





S114 · impliesBefore

interval value139 trial divisor9:natural dividend=139;positive divisor=9

After

139=15·9+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, S113





S115 · impliesBefore

interval value139 trial divisor10:natural dividend=139;positive divisor=10

After

139=13·10+9

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, S114





S116 · impliesBefore

interval value139 trial divisor11:natural dividend=139;positive divisor=11

After

139=12·11+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, S115





S117 · impliesBefore

phase=interval value139;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8, 9, 10, 11];sqrt floor=11

After

least factor at least2=139;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, S116





S118 · impliesBefore

phase=interval value140;n=140

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, S117





S119 · impliesBefore

interval value140 trial divisor2:natural dividend=140;positive divisor=2

After

140=70·2+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, S118





S120 · impliesBefore

phase=interval value140;tried ascending divisors=[2];sqrt floor=11

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, S119





S121 · impliesBefore

inclusive interval=[120, 140]

After

all primes=[127, 131, 137, 139];count=4

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, S120









## Reverse checks


- S121 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S119 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S115 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S114 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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

(120, 140, [127, 131, 137, 139], 4)

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: 274c4926be047c9ef5686d8300e2349ae45e24f41d35c0454ab218f34130e6c6
- candidate record SHA-256: 244f19890a8221ef817e7c9409c5ce17caa91cdeda7793aeea43b23677df6f7f
- 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-0100.json)