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← Question derivation index · Original question batch
R0017-09174 · q-20261013-advance-20261018-r04-prime-number-0099
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=115, hi=135, 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=115, hi=135, with lo≤hi and a bounded window. Give (lo, hi, every prime in [lo,hi] in increasing order, count).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.
Before
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.
Rule: prime_complete_evidence_scope · Uses: G001
Before
phase=interval value115;n=115
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.
Rule: prime_lower_bound_guard · Uses: G001, S001
Before
interval value115 trial divisor2:natural dividend=115;positive divisor=2
After
115=57·2+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S002
Before
interval value115 trial divisor3:natural dividend=115;positive divisor=3
After
115=38·3+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S003
Before
interval value115 trial divisor4:natural dividend=115;positive divisor=4
After
115=28·4+3
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S004
Before
interval value115 trial divisor5:natural dividend=115;positive divisor=5
After
115=23·5+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S005
Before
phase=interval value115;tried ascending divisors=[2, 3, 4, 5];sqrt floor=10
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.
Rule: complete_small_factor_bound · Uses: G001, S006
Before
phase=interval value116;n=116
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.
Rule: prime_lower_bound_guard · Uses: G001, S007
Before
interval value116 trial divisor2:natural dividend=116;positive divisor=2
After
116=58·2+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S008
Before
phase=interval value116;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.
Rule: complete_small_factor_bound · Uses: G001, S009
Before
phase=interval value117;n=117
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.
Rule: prime_lower_bound_guard · Uses: G001, S010
Before
interval value117 trial divisor2:natural dividend=117;positive divisor=2
After
117=58·2+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S011
Before
interval value117 trial divisor3:natural dividend=117;positive divisor=3
After
117=39·3+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S012
Before
phase=interval value117;tried ascending divisors=[2, 3];sqrt floor=10
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.
Rule: complete_small_factor_bound · Uses: G001, S013
Before
phase=interval value118;n=118
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.
Rule: prime_lower_bound_guard · Uses: G001, S014
Before
interval value118 trial divisor2:natural dividend=118;positive divisor=2
After
118=59·2+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S015
Before
phase=interval value118;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.
Rule: complete_small_factor_bound · Uses: G001, S016
Before
phase=interval value119;n=119
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.
Rule: prime_lower_bound_guard · Uses: G001, S017
Before
interval value119 trial divisor2:natural dividend=119;positive divisor=2
After
119=59·2+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S018
Before
interval value119 trial divisor3:natural dividend=119;positive divisor=3
After
119=39·3+2
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S019
Before
interval value119 trial divisor4:natural dividend=119;positive divisor=4
After
119=29·4+3
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S020
Before
interval value119 trial divisor5:natural dividend=119;positive divisor=5
After
119=23·5+4
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S021
Before
interval value119 trial divisor6:natural dividend=119;positive divisor=6
After
119=19·6+5
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S022
Before
interval value119 trial divisor7:natural dividend=119;positive divisor=7
After
119=17·7+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S023
Before
phase=interval value119;tried ascending divisors=[2, 3, 4, 5, 6, 7];sqrt floor=10
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.
Rule: complete_small_factor_bound · Uses: G001, S024
Before
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.
Rule: prime_lower_bound_guard · Uses: G001, S025
Before
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.
Rule: positive_euclidean_division · Uses: G001, S026
Before
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.
Rule: complete_small_factor_bound · Uses: G001, S027
Before
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.
Rule: prime_lower_bound_guard · Uses: G001, S028
Before
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.
Rule: positive_euclidean_division · Uses: G001, S029
Before
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.
Rule: positive_euclidean_division · Uses: G001, S030
Before
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.
Rule: positive_euclidean_division · Uses: G001, S031
Before
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.
Rule: positive_euclidean_division · Uses: G001, S032
Before
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.
Rule: positive_euclidean_division · Uses: G001, S033
Before
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.
Rule: positive_euclidean_division · Uses: G001, S034
Before
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.
Rule: positive_euclidean_division · Uses: G001, S035
Before
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.
Rule: positive_euclidean_division · Uses: G001, S036
Before
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.
Rule: positive_euclidean_division · Uses: G001, S037
Before
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.
Rule: positive_euclidean_division · Uses: G001, S038
Before
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.
Rule: complete_small_factor_bound · Uses: G001, S039
Before
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.
Rule: prime_lower_bound_guard · Uses: G001, S040
Before
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.
Rule: positive_euclidean_division · Uses: G001, S041
Before
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.
Rule: complete_small_factor_bound · Uses: G001, S042
Before
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.
Rule: prime_lower_bound_guard · Uses: G001, S043
Before
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.
Rule: positive_euclidean_division · Uses: G001, S044
Before
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.
Rule: positive_euclidean_division · Uses: G001, S045
Before
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.
Rule: complete_small_factor_bound · Uses: G001, S046
Before
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.
Rule: prime_lower_bound_guard · Uses: G001, S047
Before
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.
Rule: positive_euclidean_division · Uses: G001, S048
Before
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.
Rule: complete_small_factor_bound · Uses: G001, S049
Before
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.
Rule: prime_lower_bound_guard · Uses: G001, S050
Before
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.
Rule: positive_euclidean_division · Uses: G001, S051
Before
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.
Rule: positive_euclidean_division · Uses: G001, S052
Before
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.
Rule: positive_euclidean_division · Uses: G001, S053
Before
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.
Rule: positive_euclidean_division · Uses: G001, S054
Before
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.
Rule: complete_small_factor_bound · Uses: G001, S055
Before
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.
Rule: prime_lower_bound_guard · Uses: G001, S056
Before
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.
Rule: positive_euclidean_division · Uses: G001, S057
Before
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.
Rule: complete_small_factor_bound · Uses: G001, S058
Before
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.
Rule: prime_lower_bound_guard · Uses: G001, S059
Before
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.
Rule: positive_euclidean_division · Uses: G001, S060
Before
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.
Rule: positive_euclidean_division · Uses: G001, S061
Before
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.
Rule: positive_euclidean_division · Uses: G001, S062
Before
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.
Rule: positive_euclidean_division · Uses: G001, S063
Before
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.
Rule: positive_euclidean_division · Uses: G001, S064
Before
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.
Rule: positive_euclidean_division · Uses: G001, S065
Before
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.
Rule: positive_euclidean_division · Uses: G001, S066
Before
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.
Rule: positive_euclidean_division · Uses: G001, S067
Before
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.
Rule: positive_euclidean_division · Uses: G001, S068
Before
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.
Rule: positive_euclidean_division · Uses: G001, S069
Before
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.
Rule: complete_small_factor_bound · Uses: G001, S070
Before
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.
Rule: prime_lower_bound_guard · Uses: G001, S071
Before
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.
Rule: positive_euclidean_division · Uses: G001, S072
Before
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.
Rule: complete_small_factor_bound · Uses: G001, S073
Before
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.
Rule: prime_lower_bound_guard · Uses: G001, S074
Before
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.
Rule: positive_euclidean_division · Uses: G001, S075
Before
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.
Rule: positive_euclidean_division · Uses: G001, S076
Before
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.
Rule: complete_small_factor_bound · Uses: G001, S077
Before
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.
Rule: prime_lower_bound_guard · Uses: G001, S078
Before
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.
Rule: positive_euclidean_division · Uses: G001, S079
Before
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.
Rule: complete_small_factor_bound · Uses: G001, S080
Before
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.
Rule: prime_lower_bound_guard · Uses: G001, S081
Before
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.
Rule: positive_euclidean_division · Uses: G001, S082
Before
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.
Rule: positive_euclidean_division · Uses: G001, S083
Before
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.
Rule: positive_euclidean_division · Uses: G001, S084
Before
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.
Rule: positive_euclidean_division · Uses: G001, S085
Before
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.
Rule: positive_euclidean_division · Uses: G001, S086
Before
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.
Rule: positive_euclidean_division · Uses: G001, S087
Before
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.
Rule: positive_euclidean_division · Uses: G001, S088
Before
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.
Rule: positive_euclidean_division · Uses: G001, S089
Before
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.
Rule: positive_euclidean_division · Uses: G001, S090
Before
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.
Rule: positive_euclidean_division · Uses: G001, S091
Before
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.
Rule: complete_small_factor_bound · Uses: G001, S092
Before
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.
Rule: prime_lower_bound_guard · Uses: G001, S093
Before
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.
Rule: positive_euclidean_division · Uses: G001, S094
Before
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.
Rule: complete_small_factor_bound · Uses: G001, S095
Before
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.
Rule: prime_lower_bound_guard · Uses: G001, S096
Before
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.
Rule: positive_euclidean_division · Uses: G001, S097
Before
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.
Rule: positive_euclidean_division · Uses: G001, S098
Before
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.
Rule: positive_euclidean_division · Uses: G001, S099
Before
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.
Rule: positive_euclidean_division · Uses: G001, S100
Before
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.
Rule: positive_euclidean_division · Uses: G001, S101
Before
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.
Rule: positive_euclidean_division · Uses: G001, S102
Before
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.
Rule: complete_small_factor_bound · Uses: G001, S103
Before
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.
Rule: prime_lower_bound_guard · Uses: G001, S104
Before
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.
Rule: positive_euclidean_division · Uses: G001, S105
Before
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.
Rule: complete_small_factor_bound · Uses: G001, S106
Before
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.
Rule: prime_lower_bound_guard · Uses: G001, S107
Before
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.
Rule: positive_euclidean_division · Uses: G001, S108
Before
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.
Rule: positive_euclidean_division · Uses: G001, S109
Before
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.
Rule: complete_small_factor_bound · Uses: G001, S110
Before
inclusive interval=[115, 135]
After
all primes=[127, 131];count=2
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.
Rule: complete_inclusive_prime_interval · Uses: G001, S111
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.(115, 135, [127, 131], 2)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: feccdde06a115f32273e2fd635e2a98fad7ef645cc4e700086979c38afe9cddccandidate record SHA-256: 82b400a79dfd3d89016e25190a3e5338aecc1418d09db78aaf5decfa2febca9csemantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED