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← Question derivation index · Original question batch
R0017-09166 · q-20261013-advance-20261018-r04-prime-number-0091
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=75, hi=95, 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=75, hi=95, 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 value75;n=75
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S001
Before
interval value75 trial divisor2:natural dividend=75;positive divisor=2
After
75=37·2+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S002
Before
interval value75 trial divisor3:natural dividend=75;positive divisor=3
After
75=25·3+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S003
Before
phase=interval value75;tried ascending divisors=[2, 3];sqrt floor=8
After
least factor at least2=3;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S004
Before
phase=interval value76;n=76
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S005
Before
interval value76 trial divisor2:natural dividend=76;positive divisor=2
After
76=38·2+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S006
Before
phase=interval value76;tried ascending divisors=[2];sqrt floor=8
After
least factor at least2=2;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S007
Before
phase=interval value77;n=77
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S008
Before
interval value77 trial divisor2:natural dividend=77;positive divisor=2
After
77=38·2+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S009
Before
interval value77 trial divisor3:natural dividend=77;positive divisor=3
After
77=25·3+2
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S010
Before
interval value77 trial divisor4:natural dividend=77;positive divisor=4
After
77=19·4+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S011
Before
interval value77 trial divisor5:natural dividend=77;positive divisor=5
After
77=15·5+2
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S012
Before
interval value77 trial divisor6:natural dividend=77;positive divisor=6
After
77=12·6+5
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S013
Before
interval value77 trial divisor7:natural dividend=77;positive divisor=7
After
77=11·7+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S014
Before
phase=interval value77;tried ascending divisors=[2, 3, 4, 5, 6, 7];sqrt floor=8
After
least factor at least2=7;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S015
Before
phase=interval value78;n=78
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S016
Before
interval value78 trial divisor2:natural dividend=78;positive divisor=2
After
78=39·2+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S017
Before
phase=interval value78;tried ascending divisors=[2];sqrt floor=8
After
least factor at least2=2;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S018
Before
phase=interval value79;n=79
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S019
Before
interval value79 trial divisor2:natural dividend=79;positive divisor=2
After
79=39·2+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S020
Before
interval value79 trial divisor3:natural dividend=79;positive divisor=3
After
79=26·3+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S021
Before
interval value79 trial divisor4:natural dividend=79;positive divisor=4
After
79=19·4+3
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S022
Before
interval value79 trial divisor5:natural dividend=79;positive divisor=5
After
79=15·5+4
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S023
Before
interval value79 trial divisor6:natural dividend=79;positive divisor=6
After
79=13·6+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S024
Before
interval value79 trial divisor7:natural dividend=79;positive divisor=7
After
79=11·7+2
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S025
Before
interval value79 trial divisor8:natural dividend=79;positive divisor=8
After
79=9·8+7
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S026
Before
phase=interval value79;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8];sqrt floor=8
After
least factor at least2=79;prime=Yes
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S027
Before
phase=interval value80;n=80
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S028
Before
interval value80 trial divisor2:natural dividend=80;positive divisor=2
After
80=40·2+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S029
Before
phase=interval value80;tried ascending divisors=[2];sqrt floor=8
After
least factor at least2=2;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S030
Before
phase=interval value81;n=81
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S031
Before
interval value81 trial divisor2:natural dividend=81;positive divisor=2
After
81=40·2+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S032
Before
interval value81 trial divisor3:natural dividend=81;positive divisor=3
After
81=27·3+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S033
Before
phase=interval value81;tried ascending divisors=[2, 3];sqrt floor=9
After
least factor at least2=3;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S034
Before
phase=interval value82;n=82
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S035
Before
interval value82 trial divisor2:natural dividend=82;positive divisor=2
After
82=41·2+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S036
Before
phase=interval value82;tried ascending divisors=[2];sqrt floor=9
After
least factor at least2=2;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S037
Before
phase=interval value83;n=83
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S038
Before
interval value83 trial divisor2:natural dividend=83;positive divisor=2
After
83=41·2+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S039
Before
interval value83 trial divisor3:natural dividend=83;positive divisor=3
After
83=27·3+2
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S040
Before
interval value83 trial divisor4:natural dividend=83;positive divisor=4
After
83=20·4+3
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S041
Before
interval value83 trial divisor5:natural dividend=83;positive divisor=5
After
83=16·5+3
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S042
Before
interval value83 trial divisor6:natural dividend=83;positive divisor=6
After
83=13·6+5
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S043
Before
interval value83 trial divisor7:natural dividend=83;positive divisor=7
After
83=11·7+6
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S044
Before
interval value83 trial divisor8:natural dividend=83;positive divisor=8
After
83=10·8+3
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S045
Before
interval value83 trial divisor9:natural dividend=83;positive divisor=9
After
83=9·9+2
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S046
Before
phase=interval value83;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8, 9];sqrt floor=9
After
least factor at least2=83;prime=Yes
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S047
Before
phase=interval value84;n=84
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S048
Before
interval value84 trial divisor2:natural dividend=84;positive divisor=2
After
84=42·2+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S049
Before
phase=interval value84;tried ascending divisors=[2];sqrt floor=9
After
least factor at least2=2;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S050
Before
phase=interval value85;n=85
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S051
Before
interval value85 trial divisor2:natural dividend=85;positive divisor=2
After
85=42·2+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S052
Before
interval value85 trial divisor3:natural dividend=85;positive divisor=3
After
85=28·3+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S053
Before
interval value85 trial divisor4:natural dividend=85;positive divisor=4
After
85=21·4+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S054
Before
interval value85 trial divisor5:natural dividend=85;positive divisor=5
After
85=17·5+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S055
Before
phase=interval value85;tried ascending divisors=[2, 3, 4, 5];sqrt floor=9
After
least factor at least2=5;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S056
Before
phase=interval value86;n=86
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, S057
Before
interval value86 trial divisor2:natural dividend=86;positive divisor=2
After
86=43·2+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, S058
Before
phase=interval value86;tried ascending divisors=[2];sqrt floor=9
After
least factor at least2=2;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S059
Before
phase=interval value87;n=87
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, S060
Before
interval value87 trial divisor2:natural dividend=87;positive divisor=2
After
87=43·2+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, S061
Before
interval value87 trial divisor3:natural dividend=87;positive divisor=3
After
87=29·3+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, S062
Before
phase=interval value87;tried ascending divisors=[2, 3];sqrt floor=9
After
least factor at least2=3;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S063
Before
phase=interval value88;n=88
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, S064
Before
interval value88 trial divisor2:natural dividend=88;positive divisor=2
After
88=44·2+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, S065
Before
phase=interval value88;tried ascending divisors=[2];sqrt floor=9
After
least factor at least2=2;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S066
Before
phase=interval value89;n=89
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, S067
Before
interval value89 trial divisor2:natural dividend=89;positive divisor=2
After
89=44·2+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, S068
Before
interval value89 trial divisor3:natural dividend=89;positive divisor=3
After
89=29·3+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, S069
Before
interval value89 trial divisor4:natural dividend=89;positive divisor=4
After
89=22·4+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, S070
Before
interval value89 trial divisor5:natural dividend=89;positive divisor=5
After
89=17·5+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, S071
Before
interval value89 trial divisor6:natural dividend=89;positive divisor=6
After
89=14·6+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, S072
Before
interval value89 trial divisor7:natural dividend=89;positive divisor=7
After
89=12·7+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, S073
Before
interval value89 trial divisor8:natural dividend=89;positive divisor=8
After
89=11·8+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, S074
Before
interval value89 trial divisor9:natural dividend=89;positive divisor=9
After
89=9·9+8
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
phase=interval value89;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8, 9];sqrt floor=9
After
least factor at least2=89;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, S076
Before
phase=interval value90;n=90
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, S077
Before
interval value90 trial divisor2:natural dividend=90;positive divisor=2
After
90=45·2+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, S078
Before
phase=interval value90;tried ascending divisors=[2];sqrt floor=9
After
least factor at least2=2;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S079
Before
phase=interval value91;n=91
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, S080
Before
interval value91 trial divisor2:natural dividend=91;positive divisor=2
After
91=45·2+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, S081
Before
interval value91 trial divisor3:natural dividend=91;positive divisor=3
After
91=30·3+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, S082
Before
interval value91 trial divisor4:natural dividend=91;positive divisor=4
After
91=22·4+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, S083
Before
interval value91 trial divisor5:natural dividend=91;positive divisor=5
After
91=18·5+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, S084
Before
interval value91 trial divisor6:natural dividend=91;positive divisor=6
After
91=15·6+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, S085
Before
interval value91 trial divisor7:natural dividend=91;positive divisor=7
After
91=13·7+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, S086
Before
phase=interval value91;tried ascending divisors=[2, 3, 4, 5, 6, 7];sqrt floor=9
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, S087
Before
phase=interval value92;n=92
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, S088
Before
interval value92 trial divisor2:natural dividend=92;positive divisor=2
After
92=46·2+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, S089
Before
phase=interval value92;tried ascending divisors=[2];sqrt floor=9
After
least factor at least2=2;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S090
Before
phase=interval value93;n=93
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, S091
Before
interval value93 trial divisor2:natural dividend=93;positive divisor=2
After
93=46·2+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, S092
Before
interval value93 trial divisor3:natural dividend=93;positive divisor=3
After
93=31·3+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, S093
Before
phase=interval value93;tried ascending divisors=[2, 3];sqrt floor=9
After
least factor at least2=3;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S094
Before
phase=interval value94;n=94
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, S095
Before
interval value94 trial divisor2:natural dividend=94;positive divisor=2
After
94=47·2+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, S096
Before
phase=interval value94;tried ascending divisors=[2];sqrt floor=9
After
least factor at least2=2;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S097
Before
phase=interval value95;n=95
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, S098
Before
interval value95 trial divisor2:natural dividend=95;positive divisor=2
After
95=47·2+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, S099
Before
interval value95 trial divisor3:natural dividend=95;positive divisor=3
After
95=31·3+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, S100
Before
interval value95 trial divisor4:natural dividend=95;positive divisor=4
After
95=23·4+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, S101
Before
interval value95 trial divisor5:natural dividend=95;positive divisor=5
After
95=19·5+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, S102
Before
phase=interval value95;tried ascending divisors=[2, 3, 4, 5];sqrt floor=9
After
least factor at least2=5;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S103
Before
inclusive interval=[75, 95]
After
all primes=[79, 83, 89];count=3
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, S104
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.(75, 95, [79, 83, 89], 3)
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: 5924d3c7dbf304c9be39683a4532cc5d7dbee2467b964fc6b1beae163ab94661candidate record SHA-256: e361469173a970798dc30378c1c4d39e950b1948f5b535740a8107a2240e268csemantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED