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← Question derivation index · Original question batch
R0017-09156 · q-20261013-advance-20261018-r04-prime-number-0081
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=25, hi=45, 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=25, hi=45, 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 value25;n=25
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 value25 trial divisor2:natural dividend=25;positive divisor=2
After
25=12·2+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 value25 trial divisor3:natural dividend=25;positive divisor=3
After
25=8·3+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 value25 trial divisor4:natural dividend=25;positive divisor=4
After
25=6·4+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, S004
Before
interval value25 trial divisor5:natural dividend=25;positive divisor=5
After
25=5·5+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 value25;tried ascending divisors=[2, 3, 4, 5];sqrt floor=5
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 value26;n=26
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 value26 trial divisor2:natural dividend=26;positive divisor=2
After
26=13·2+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 value26;tried ascending divisors=[2];sqrt floor=5
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 value27;n=27
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 value27 trial divisor2:natural dividend=27;positive divisor=2
After
27=13·2+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 value27 trial divisor3:natural dividend=27;positive divisor=3
After
27=9·3+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 value27;tried ascending divisors=[2, 3];sqrt floor=5
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 value28;n=28
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 value28 trial divisor2:natural dividend=28;positive divisor=2
After
28=14·2+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 value28;tried ascending divisors=[2];sqrt floor=5
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 value29;n=29
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 value29 trial divisor2:natural dividend=29;positive divisor=2
After
29=14·2+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 value29 trial divisor3:natural dividend=29;positive divisor=3
After
29=9·3+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 value29 trial divisor4:natural dividend=29;positive divisor=4
After
29=7·4+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, S020
Before
interval value29 trial divisor5:natural dividend=29;positive divisor=5
After
29=5·5+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
phase=interval value29;tried ascending divisors=[2, 3, 4, 5];sqrt floor=5
After
least factor at least2=29;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, S022
Before
phase=interval value30;n=30
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, S023
Before
interval value30 trial divisor2:natural dividend=30;positive divisor=2
After
30=15·2+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, S024
Before
phase=interval value30;tried ascending divisors=[2];sqrt floor=5
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, S025
Before
phase=interval value31;n=31
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, S026
Before
interval value31 trial divisor2:natural dividend=31;positive divisor=2
After
31=15·2+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, S027
Before
interval value31 trial divisor3:natural dividend=31;positive divisor=3
After
31=10·3+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, S028
Before
interval value31 trial divisor4:natural dividend=31;positive divisor=4
After
31=7·4+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, S029
Before
interval value31 trial divisor5:natural dividend=31;positive divisor=5
After
31=6·5+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, S030
Before
phase=interval value31;tried ascending divisors=[2, 3, 4, 5];sqrt floor=5
After
least factor at least2=31;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, S031
Before
phase=interval value32;n=32
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, S032
Before
interval value32 trial divisor2:natural dividend=32;positive divisor=2
After
32=16·2+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, S033
Before
phase=interval value32;tried ascending divisors=[2];sqrt floor=5
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, S034
Before
phase=interval value33;n=33
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 value33 trial divisor2:natural dividend=33;positive divisor=2
After
33=16·2+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, S036
Before
interval value33 trial divisor3:natural dividend=33;positive divisor=3
After
33=11·3+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, S037
Before
phase=interval value33;tried ascending divisors=[2, 3];sqrt floor=5
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, S038
Before
phase=interval value34;n=34
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 value34 trial divisor2:natural dividend=34;positive divisor=2
After
34=17·2+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, S040
Before
phase=interval value34;tried ascending divisors=[2];sqrt floor=5
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, S041
Before
phase=interval value35;n=35
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, S042
Before
interval value35 trial divisor2:natural dividend=35;positive divisor=2
After
35=17·2+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, S043
Before
interval value35 trial divisor3:natural dividend=35;positive divisor=3
After
35=11·3+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, S044
Before
interval value35 trial divisor4:natural dividend=35;positive divisor=4
After
35=8·4+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, S045
Before
interval value35 trial divisor5:natural dividend=35;positive divisor=5
After
35=7·5+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, S046
Before
phase=interval value35;tried ascending divisors=[2, 3, 4, 5];sqrt floor=5
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, S047
Before
phase=interval value36;n=36
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 value36 trial divisor2:natural dividend=36;positive divisor=2
After
36=18·2+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 value36;tried ascending divisors=[2];sqrt floor=6
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 value37;n=37
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 value37 trial divisor2:natural dividend=37;positive divisor=2
After
37=18·2+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 value37 trial divisor3:natural dividend=37;positive divisor=3
After
37=12·3+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 value37 trial divisor4:natural dividend=37;positive divisor=4
After
37=9·4+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 value37 trial divisor5:natural dividend=37;positive divisor=5
After
37=7·5+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, S055
Before
interval value37 trial divisor6:natural dividend=37;positive divisor=6
After
37=6·6+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, S056
Before
phase=interval value37;tried ascending divisors=[2, 3, 4, 5, 6];sqrt floor=6
After
least factor at least2=37;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, S057
Before
phase=interval value38;n=38
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, S058
Before
interval value38 trial divisor2:natural dividend=38;positive divisor=2
After
38=19·2+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, S059
Before
phase=interval value38;tried ascending divisors=[2];sqrt floor=6
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, S060
Before
phase=interval value39;n=39
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, S061
Before
interval value39 trial divisor2:natural dividend=39;positive divisor=2
After
39=19·2+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, S062
Before
interval value39 trial divisor3:natural dividend=39;positive divisor=3
After
39=13·3+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, S063
Before
phase=interval value39;tried ascending divisors=[2, 3];sqrt floor=6
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, S064
Before
phase=interval value40;n=40
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, S065
Before
interval value40 trial divisor2:natural dividend=40;positive divisor=2
After
40=20·2+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, S066
Before
phase=interval value40;tried ascending divisors=[2];sqrt floor=6
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, S067
Before
phase=interval value41;n=41
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, S068
Before
interval value41 trial divisor2:natural dividend=41;positive divisor=2
After
41=20·2+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, S069
Before
interval value41 trial divisor3:natural dividend=41;positive divisor=3
After
41=13·3+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, S070
Before
interval value41 trial divisor4:natural dividend=41;positive divisor=4
After
41=10·4+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, S071
Before
interval value41 trial divisor5:natural dividend=41;positive divisor=5
After
41=8·5+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, S072
Before
interval value41 trial divisor6:natural dividend=41;positive divisor=6
After
41=6·6+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, S073
Before
phase=interval value41;tried ascending divisors=[2, 3, 4, 5, 6];sqrt floor=6
After
least factor at least2=41;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, S074
Before
phase=interval value42;n=42
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, S075
Before
interval value42 trial divisor2:natural dividend=42;positive divisor=2
After
42=21·2+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, S076
Before
phase=interval value42;tried ascending divisors=[2];sqrt floor=6
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, S077
Before
phase=interval value43;n=43
After
n>=2=Yes
Conditions: none
A prime must be at least2.Exclude0 and1 first rather than treating an absence of trial factors as primality.
Rule: prime_lower_bound_guard · Uses: G001, S078
Before
interval value43 trial divisor2:natural dividend=43;positive divisor=2
After
43=21·2+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, S079
Before
interval value43 trial divisor3:natural dividend=43;positive divisor=3
After
43=14·3+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, S080
Before
interval value43 trial divisor4:natural dividend=43;positive divisor=4
After
43=10·4+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, S081
Before
interval value43 trial divisor5:natural dividend=43;positive divisor=5
After
43=8·5+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, S082
Before
interval value43 trial divisor6:natural dividend=43;positive divisor=6
After
43=7·6+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, S083
Before
phase=interval value43;tried ascending divisors=[2, 3, 4, 5, 6];sqrt floor=6
After
least factor at least2=43;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, S084
Before
phase=interval value44;n=44
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, S085
Before
interval value44 trial divisor2:natural dividend=44;positive divisor=2
After
44=22·2+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, S086
Before
phase=interval value44;tried ascending divisors=[2];sqrt floor=6
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, S087
Before
phase=interval value45;n=45
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 value45 trial divisor2:natural dividend=45;positive divisor=2
After
45=22·2+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, S089
Before
interval value45 trial divisor3:natural dividend=45;positive divisor=3
After
45=15·3+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, S090
Before
phase=interval value45;tried ascending divisors=[2, 3];sqrt floor=6
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, S091
Before
inclusive interval=[25, 45]
After
all primes=[29, 31, 37, 41, 43];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, S092
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.(25, 45, [29, 31, 37, 41, 43], 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: f62625a8f220a3b656dca3a92fe9d9daa260326903bd934be745000071606f42candidate record SHA-256: afca78291329b31f1d0fe9ce1adb86fdebe1d0faf21c90b84e333e0b476f131bsemantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED