Step |
Hyp |
Ref |
Expression |
1 |
|
fmtnonn |
⊢ ( 𝑁 ∈ ℕ0 → ( FermatNo ‘ 𝑁 ) ∈ ℕ ) |
2 |
1
|
nnzd |
⊢ ( 𝑁 ∈ ℕ0 → ( FermatNo ‘ 𝑁 ) ∈ ℤ ) |
3 |
|
fmtnonn |
⊢ ( 𝑀 ∈ ℕ0 → ( FermatNo ‘ 𝑀 ) ∈ ℕ ) |
4 |
3
|
nnzd |
⊢ ( 𝑀 ∈ ℕ0 → ( FermatNo ‘ 𝑀 ) ∈ ℤ ) |
5 |
2 4
|
anim12ci |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) → ( ( FermatNo ‘ 𝑀 ) ∈ ℤ ∧ ( FermatNo ‘ 𝑁 ) ∈ ℤ ) ) |
6 |
5
|
3adant3 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( FermatNo ‘ 𝑀 ) ∈ ℤ ∧ ( FermatNo ‘ 𝑁 ) ∈ ℤ ) ) |
7 |
|
gcddvds |
⊢ ( ( ( FermatNo ‘ 𝑀 ) ∈ ℤ ∧ ( FermatNo ‘ 𝑁 ) ∈ ℤ ) → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑀 ) ∧ ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑁 ) ) ) |
8 |
6 7
|
syl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑀 ) ∧ ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑁 ) ) ) |
9 |
|
goldbachthlem1 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( FermatNo ‘ 𝑀 ) ∥ ( ( FermatNo ‘ 𝑁 ) − 2 ) ) |
10 |
|
gcdcl |
⊢ ( ( ( FermatNo ‘ 𝑀 ) ∈ ℤ ∧ ( FermatNo ‘ 𝑁 ) ∈ ℤ ) → ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∈ ℕ0 ) |
11 |
6 10
|
syl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∈ ℕ0 ) |
12 |
11
|
nn0zd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∈ ℤ ) |
13 |
4
|
3ad2ant2 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( FermatNo ‘ 𝑀 ) ∈ ℤ ) |
14 |
|
2z |
⊢ 2 ∈ ℤ |
15 |
14
|
a1i |
⊢ ( 𝑁 ∈ ℕ0 → 2 ∈ ℤ ) |
16 |
2 15
|
zsubcld |
⊢ ( 𝑁 ∈ ℕ0 → ( ( FermatNo ‘ 𝑁 ) − 2 ) ∈ ℤ ) |
17 |
16
|
3ad2ant1 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( FermatNo ‘ 𝑁 ) − 2 ) ∈ ℤ ) |
18 |
|
dvdstr |
⊢ ( ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∈ ℤ ∧ ( FermatNo ‘ 𝑀 ) ∈ ℤ ∧ ( ( FermatNo ‘ 𝑁 ) − 2 ) ∈ ℤ ) → ( ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑀 ) ∧ ( FermatNo ‘ 𝑀 ) ∥ ( ( FermatNo ‘ 𝑁 ) − 2 ) ) → ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( ( FermatNo ‘ 𝑁 ) − 2 ) ) ) |
19 |
12 13 17 18
|
syl3anc |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑀 ) ∧ ( FermatNo ‘ 𝑀 ) ∥ ( ( FermatNo ‘ 𝑁 ) − 2 ) ) → ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( ( FermatNo ‘ 𝑁 ) − 2 ) ) ) |
20 |
9 19
|
mpan2d |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑀 ) → ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( ( FermatNo ‘ 𝑁 ) − 2 ) ) ) |
21 |
2
|
3ad2ant1 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( FermatNo ‘ 𝑁 ) ∈ ℤ ) |
22 |
|
dvds2sub |
⊢ ( ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∈ ℤ ∧ ( FermatNo ‘ 𝑁 ) ∈ ℤ ∧ ( ( FermatNo ‘ 𝑁 ) − 2 ) ∈ ℤ ) → ( ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑁 ) ∧ ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( ( FermatNo ‘ 𝑁 ) − 2 ) ) → ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( ( FermatNo ‘ 𝑁 ) − ( ( FermatNo ‘ 𝑁 ) − 2 ) ) ) ) |
23 |
12 21 17 22
|
syl3anc |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑁 ) ∧ ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( ( FermatNo ‘ 𝑁 ) − 2 ) ) → ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( ( FermatNo ‘ 𝑁 ) − ( ( FermatNo ‘ 𝑁 ) − 2 ) ) ) ) |
24 |
23
|
ancomsd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( ( FermatNo ‘ 𝑁 ) − 2 ) ∧ ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑁 ) ) → ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( ( FermatNo ‘ 𝑁 ) − ( ( FermatNo ‘ 𝑁 ) − 2 ) ) ) ) |
25 |
1
|
nncnd |
⊢ ( 𝑁 ∈ ℕ0 → ( FermatNo ‘ 𝑁 ) ∈ ℂ ) |
26 |
25
|
3ad2ant1 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( FermatNo ‘ 𝑁 ) ∈ ℂ ) |
27 |
|
2cnd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → 2 ∈ ℂ ) |
28 |
26 27
|
nncand |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( FermatNo ‘ 𝑁 ) − ( ( FermatNo ‘ 𝑁 ) − 2 ) ) = 2 ) |
29 |
28
|
breq2d |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( ( FermatNo ‘ 𝑁 ) − ( ( FermatNo ‘ 𝑁 ) − 2 ) ) ↔ ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ 2 ) ) |
30 |
|
2prm |
⊢ 2 ∈ ℙ |
31 |
1 3
|
anim12ci |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) → ( ( FermatNo ‘ 𝑀 ) ∈ ℕ ∧ ( FermatNo ‘ 𝑁 ) ∈ ℕ ) ) |
32 |
31
|
3adant3 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( FermatNo ‘ 𝑀 ) ∈ ℕ ∧ ( FermatNo ‘ 𝑁 ) ∈ ℕ ) ) |
33 |
|
gcdnncl |
⊢ ( ( ( FermatNo ‘ 𝑀 ) ∈ ℕ ∧ ( FermatNo ‘ 𝑁 ) ∈ ℕ ) → ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∈ ℕ ) |
34 |
32 33
|
syl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∈ ℕ ) |
35 |
|
dvdsprime |
⊢ ( ( 2 ∈ ℙ ∧ ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∈ ℕ ) → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ 2 ↔ ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) = 2 ∨ ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) = 1 ) ) ) |
36 |
30 34 35
|
sylancr |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ 2 ↔ ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) = 2 ∨ ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) = 1 ) ) ) |
37 |
5 7
|
syl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑀 ) ∧ ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑁 ) ) ) |
38 |
|
breq1 |
⊢ ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) = 2 → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑁 ) ↔ 2 ∥ ( FermatNo ‘ 𝑁 ) ) ) |
39 |
38
|
adantl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) ∧ ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) = 2 ) → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑁 ) ↔ 2 ∥ ( FermatNo ‘ 𝑁 ) ) ) |
40 |
|
fmtnoodd |
⊢ ( 𝑁 ∈ ℕ0 → ¬ 2 ∥ ( FermatNo ‘ 𝑁 ) ) |
41 |
40
|
pm2.21d |
⊢ ( 𝑁 ∈ ℕ0 → ( 2 ∥ ( FermatNo ‘ 𝑁 ) → ( ( FermatNo ‘ 𝑁 ) gcd ( FermatNo ‘ 𝑀 ) ) = 1 ) ) |
42 |
41
|
ad2antrr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) ∧ ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) = 2 ) → ( 2 ∥ ( FermatNo ‘ 𝑁 ) → ( ( FermatNo ‘ 𝑁 ) gcd ( FermatNo ‘ 𝑀 ) ) = 1 ) ) |
43 |
39 42
|
sylbid |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) ∧ ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) = 2 ) → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑁 ) → ( ( FermatNo ‘ 𝑁 ) gcd ( FermatNo ‘ 𝑀 ) ) = 1 ) ) |
44 |
43
|
ex |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) = 2 → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑁 ) → ( ( FermatNo ‘ 𝑁 ) gcd ( FermatNo ‘ 𝑀 ) ) = 1 ) ) ) |
45 |
44
|
com23 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑁 ) → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) = 2 → ( ( FermatNo ‘ 𝑁 ) gcd ( FermatNo ‘ 𝑀 ) ) = 1 ) ) ) |
46 |
45
|
adantld |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) → ( ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑀 ) ∧ ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑁 ) ) → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) = 2 → ( ( FermatNo ‘ 𝑁 ) gcd ( FermatNo ‘ 𝑀 ) ) = 1 ) ) ) |
47 |
37 46
|
mpd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) = 2 → ( ( FermatNo ‘ 𝑁 ) gcd ( FermatNo ‘ 𝑀 ) ) = 1 ) ) |
48 |
47
|
3adant3 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) = 2 → ( ( FermatNo ‘ 𝑁 ) gcd ( FermatNo ‘ 𝑀 ) ) = 1 ) ) |
49 |
|
gcdcom |
⊢ ( ( ( FermatNo ‘ 𝑀 ) ∈ ℤ ∧ ( FermatNo ‘ 𝑁 ) ∈ ℤ ) → ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) = ( ( FermatNo ‘ 𝑁 ) gcd ( FermatNo ‘ 𝑀 ) ) ) |
50 |
6 49
|
syl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) = ( ( FermatNo ‘ 𝑁 ) gcd ( FermatNo ‘ 𝑀 ) ) ) |
51 |
50
|
eqeq1d |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) = 1 ↔ ( ( FermatNo ‘ 𝑁 ) gcd ( FermatNo ‘ 𝑀 ) ) = 1 ) ) |
52 |
51
|
biimpd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) = 1 → ( ( FermatNo ‘ 𝑁 ) gcd ( FermatNo ‘ 𝑀 ) ) = 1 ) ) |
53 |
48 52
|
jaod |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) = 2 ∨ ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) = 1 ) → ( ( FermatNo ‘ 𝑁 ) gcd ( FermatNo ‘ 𝑀 ) ) = 1 ) ) |
54 |
36 53
|
sylbid |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ 2 → ( ( FermatNo ‘ 𝑁 ) gcd ( FermatNo ‘ 𝑀 ) ) = 1 ) ) |
55 |
29 54
|
sylbid |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( ( FermatNo ‘ 𝑁 ) − ( ( FermatNo ‘ 𝑁 ) − 2 ) ) → ( ( FermatNo ‘ 𝑁 ) gcd ( FermatNo ‘ 𝑀 ) ) = 1 ) ) |
56 |
24 55
|
syld |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( ( FermatNo ‘ 𝑁 ) − 2 ) ∧ ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑁 ) ) → ( ( FermatNo ‘ 𝑁 ) gcd ( FermatNo ‘ 𝑀 ) ) = 1 ) ) |
57 |
20 56
|
syland |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑀 ) ∧ ( ( FermatNo ‘ 𝑀 ) gcd ( FermatNo ‘ 𝑁 ) ) ∥ ( FermatNo ‘ 𝑁 ) ) → ( ( FermatNo ‘ 𝑁 ) gcd ( FermatNo ‘ 𝑀 ) ) = 1 ) ) |
58 |
8 57
|
mpd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 < 𝑁 ) → ( ( FermatNo ‘ 𝑁 ) gcd ( FermatNo ‘ 𝑀 ) ) = 1 ) |