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← Question derivation index · Original question batch
R0017-09162 · q-20261013-advance-20261018-r04-prime-number-0087
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=55, hi=75, 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=55, hi=75, 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 value55;n=55
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 value55 trial divisor2:natural dividend=55;positive divisor=2
After
55=27·2+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 value55 trial divisor3:natural dividend=55;positive divisor=3
After
55=18·3+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 value55 trial divisor4:natural dividend=55;positive divisor=4
After
55=13·4+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 value55 trial divisor5:natural dividend=55;positive divisor=5
After
55=11·5+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 value55;tried ascending divisors=[2, 3, 4, 5];sqrt floor=7
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 value56;n=56
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 value56 trial divisor2:natural dividend=56;positive divisor=2
After
56=28·2+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 value56;tried ascending divisors=[2];sqrt floor=7
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 value57;n=57
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 value57 trial divisor2:natural dividend=57;positive divisor=2
After
57=28·2+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 value57 trial divisor3:natural dividend=57;positive divisor=3
After
57=19·3+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 value57;tried ascending divisors=[2, 3];sqrt floor=7
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 value58;n=58
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 value58 trial divisor2:natural dividend=58;positive divisor=2
After
58=29·2+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 value58;tried ascending divisors=[2];sqrt floor=7
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 value59;n=59
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 value59 trial divisor2:natural dividend=59;positive divisor=2
After
59=29·2+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 value59 trial divisor3:natural dividend=59;positive divisor=3
After
59=19·3+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 value59 trial divisor4:natural dividend=59;positive divisor=4
After
59=14·4+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 value59 trial divisor5:natural dividend=59;positive divisor=5
After
59=11·5+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 value59 trial divisor6:natural dividend=59;positive divisor=6
After
59=9·6+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 value59 trial divisor7:natural dividend=59;positive divisor=7
After
59=8·7+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, S023
Before
phase=interval value59;tried ascending divisors=[2, 3, 4, 5, 6, 7];sqrt floor=7
After
least factor at least2=59;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, S024
Before
phase=interval value60;n=60
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 value60 trial divisor2:natural dividend=60;positive divisor=2
After
60=30·2+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 value60;tried ascending divisors=[2];sqrt floor=7
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 value61;n=61
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 value61 trial divisor2:natural dividend=61;positive divisor=2
After
61=30·2+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 value61 trial divisor3:natural dividend=61;positive divisor=3
After
61=20·3+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 value61 trial divisor4:natural dividend=61;positive divisor=4
After
61=15·4+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 value61 trial divisor5:natural dividend=61;positive divisor=5
After
61=12·5+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 value61 trial divisor6:natural dividend=61;positive divisor=6
After
61=10·6+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 value61 trial divisor7:natural dividend=61;positive divisor=7
After
61=8·7+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, S034
Before
phase=interval value61;tried ascending divisors=[2, 3, 4, 5, 6, 7];sqrt floor=7
After
least factor at least2=61;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, S035
Before
phase=interval value62;n=62
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, S036
Before
interval value62 trial divisor2:natural dividend=62;positive divisor=2
After
62=31·2+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, S037
Before
phase=interval value62;tried ascending divisors=[2];sqrt floor=7
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, S038
Before
phase=interval value63;n=63
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, S039
Before
interval value63 trial divisor2:natural dividend=63;positive divisor=2
After
63=31·2+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, S040
Before
interval value63 trial divisor3:natural dividend=63;positive divisor=3
After
63=21·3+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, S041
Before
phase=interval value63;tried ascending divisors=[2, 3];sqrt floor=7
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, S042
Before
phase=interval value64;n=64
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 value64 trial divisor2:natural dividend=64;positive divisor=2
After
64=32·2+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, S044
Before
phase=interval value64;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, S045
Before
phase=interval value65;n=65
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S046
Before
interval value65 trial divisor2:natural dividend=65;positive divisor=2
After
65=32·2+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S047
Before
interval value65 trial divisor3:natural dividend=65;positive divisor=3
After
65=21·3+2
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S048
Before
interval value65 trial divisor4:natural dividend=65;positive divisor=4
After
65=16·4+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S049
Before
interval value65 trial divisor5:natural dividend=65;positive divisor=5
After
65=13·5+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S050
Before
phase=interval value65;tried ascending divisors=[2, 3, 4, 5];sqrt floor=8
After
least factor at least2=5;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S051
Before
phase=interval value66;n=66
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S052
Before
interval value66 trial divisor2:natural dividend=66;positive divisor=2
After
66=33·2+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S053
Before
phase=interval value66;tried ascending divisors=[2];sqrt floor=8
After
least factor at least2=2;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S054
Before
phase=interval value67;n=67
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S055
Before
interval value67 trial divisor2:natural dividend=67;positive divisor=2
After
67=33·2+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S056
Before
interval value67 trial divisor3:natural dividend=67;positive divisor=3
After
67=22·3+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S057
Before
interval value67 trial divisor4:natural dividend=67;positive divisor=4
After
67=16·4+3
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S058
Before
interval value67 trial divisor5:natural dividend=67;positive divisor=5
After
67=13·5+2
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S059
Before
interval value67 trial divisor6:natural dividend=67;positive divisor=6
After
67=11·6+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S060
Before
interval value67 trial divisor7:natural dividend=67;positive divisor=7
After
67=9·7+4
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S061
Before
interval value67 trial divisor8:natural dividend=67;positive divisor=8
After
67=8·8+3
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S062
Before
phase=interval value67;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8];sqrt floor=8
After
least factor at least2=67;prime=Yes
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S063
Before
phase=interval value68;n=68
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S064
Before
interval value68 trial divisor2:natural dividend=68;positive divisor=2
After
68=34·2+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S065
Before
phase=interval value68;tried ascending divisors=[2];sqrt floor=8
After
least factor at least2=2;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S066
Before
phase=interval value69;n=69
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S067
Before
interval value69 trial divisor2:natural dividend=69;positive divisor=2
After
69=34·2+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S068
Before
interval value69 trial divisor3:natural dividend=69;positive divisor=3
After
69=23·3+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S069
Before
phase=interval value69;tried ascending divisors=[2, 3];sqrt floor=8
After
least factor at least2=3;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S070
Before
phase=interval value70;n=70
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S071
Before
interval value70 trial divisor2:natural dividend=70;positive divisor=2
After
70=35·2+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S072
Before
phase=interval value70;tried ascending divisors=[2];sqrt floor=8
After
least factor at least2=2;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S073
Before
phase=interval value71;n=71
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S074
Before
interval value71 trial divisor2:natural dividend=71;positive divisor=2
After
71=35·2+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S075
Before
interval value71 trial divisor3:natural dividend=71;positive divisor=3
After
71=23·3+2
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S076
Before
interval value71 trial divisor4:natural dividend=71;positive divisor=4
After
71=17·4+3
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S077
Before
interval value71 trial divisor5:natural dividend=71;positive divisor=5
After
71=14·5+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S078
Before
interval value71 trial divisor6:natural dividend=71;positive divisor=6
After
71=11·6+5
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S079
Before
interval value71 trial divisor7:natural dividend=71;positive divisor=7
After
71=10·7+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S080
Before
interval value71 trial divisor8:natural dividend=71;positive divisor=8
After
71=8·8+7
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S081
Before
phase=interval value71;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8];sqrt floor=8
After
least factor at least2=71;prime=Yes
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S082
Before
phase=interval value72;n=72
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S083
Before
interval value72 trial divisor2:natural dividend=72;positive divisor=2
After
72=36·2+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S084
Before
phase=interval value72;tried ascending divisors=[2];sqrt floor=8
After
least factor at least2=2;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S085
Before
phase=interval value73;n=73
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S086
Before
interval value73 trial divisor2:natural dividend=73;positive divisor=2
After
73=36·2+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S087
Before
interval value73 trial divisor3:natural dividend=73;positive divisor=3
After
73=24·3+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S088
Before
interval value73 trial divisor4:natural dividend=73;positive divisor=4
After
73=18·4+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S089
Before
interval value73 trial divisor5:natural dividend=73;positive divisor=5
After
73=14·5+3
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S090
Before
interval value73 trial divisor6:natural dividend=73;positive divisor=6
After
73=12·6+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S091
Before
interval value73 trial divisor7:natural dividend=73;positive divisor=7
After
73=10·7+3
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S092
Before
interval value73 trial divisor8:natural dividend=73;positive divisor=8
After
73=9·8+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S093
Before
phase=interval value73;tried ascending divisors=[2, 3, 4, 5, 6, 7, 8];sqrt floor=8
After
least factor at least2=73;prime=Yes
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S094
Before
phase=interval value74;n=74
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S095
Before
interval value74 trial divisor2:natural dividend=74;positive divisor=2
After
74=37·2+0
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S096
Before
phase=interval value74;tried ascending divisors=[2];sqrt floor=8
After
least factor at least2=2;prime=No
Conditions: none
If n=uv is composite with u,v≥2,one factor is≤sqrt(n).The first zero remainder in ascending order gives the least factor;if none exists through the full bound,n itself is the least factor≥2 and is prime.
Rule: complete_small_factor_bound · Uses: G001, S097
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, S098
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, S099
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, S100
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, S101
Before
inclusive interval=[55, 75]
After
all primes=[59, 61, 67, 71, 73];count=5
Conditions: none
Check both interval endpoints and retain exactly those values passing the complete primality test.This finite enumeration makes no distribution claim outside the interval.
Rule: complete_inclusive_prime_interval · Uses: G001, S102
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.(55, 75, [59, 61, 67, 71, 73], 5)
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: 8dddc08a253c7897cee500e9be97a23b77628bb804ec02adb4c45e138b16394ccandidate record SHA-256: 3553cd9b8193d606051924303d533a15b015e014736d476c0424504da419e6e8semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED