lý thuyết số của đàm văn nhỉ

Z.
Z
Z
m
m
τ(n), σ(n)
ϕ(n)
ax ≡ b(mod m)
ax ≡ b(mod m)
Z
Z Z
+
N N
+
Q
R R
+
a, b ∈ Z, b = 0. a
b b a c ∈ Z a = bc.
a b a : b.
b|a b a. a = bc b
a.
1| a ∀a ∈ Z,
a| a ∀a ∈ Z, a = 0,
a|b, b|c a|c ∀a, b, c ∈ Z, a, b = 0,
a|b |a| ≤ |b| ∀a, b ∈ Z, a, b = 0,
a|b
i

∀a, b
i
∈ Z, i = 1, . . . , n , a|

n
i=1
b
i
x
i
x
i
∈
Z.
a|b b|a a = b a = −b ∀a, b ∈ Z, a, b = 0.
b|a Z
5|−5, −5|5, 5 = −5.
Z.
a, b ∈ Z, b = 0,
q, r ∈ Z a = qb + r, 0 ≤ r < |b|.
T = {n|b| | n|b| ≤ a, n ∈ Z}. |b| ≥ 1
−|a||b| ≤ −|a| ≤ a. −|a||b| ∈ T. T = ∅. T
T m|b|. m|b| ≤ a r = a−m|b| ≥ 0
r ∈ Z. (m + 1)|b| = m|b| + |b| > m|b|.
m|b| T (m + 1)|b| > a. |b| > a − m|b| = r
a = qb + r 0 ≤ r < |b|.
a = qb + r, 0 ≤ r < |b|,
a = q
1
b + r

1
, 0 ≤ r
1
< |b|.
r − r
1
= b(q
1
− q). |r − r
1
| < |b|
|q
1
− q||b| < |b|. q = q
1
r = r
1
.
a = qb + r, 0 ≤ r < |b|. r = 0 q
a b. r = 0 q r
a b.
57 = 14.4 + 1, (a = 57, b = 14, q = 4, r = 1).
−65 = (−17).4 + 3, (a = −65, b = 4, q = −17, r = 3).
−134 = 5.(−32) + 26, (a = −134, b = −32, q = 5, r = 26).
a
1
, . . . , a
n
∈ Z
d a

i
d|a
i
i = 1, . . . , n.
a
1
, . . . , a
n
∈ Z uc(a
1
, . . . , a
n
)
uc(18, −15, 21) = {1, −1, 3, −3}.
a
1
, . . . , a
n
∈ Z
d a
i
d a
i
d
d a
1
, . . . , a
n
∈ Z
d|a

i
, i = 1, . . . , n, c|a
i
, i = 1, . . . , n, c|d.
d a
1
, . . . , a
n
−d
a
1
, . . . , a
n
. a
1
, . . . , a
n
ucln(a
1
, . . . , a
n
) |d|. ucln(a
1
, . . . , a
n
)
uc(a
1
, . . . , a
n

).
0, 0, . . . , 0. uc(0, . . . , 0) = Z.
ucln(0, . . . , 0).
a
1
, . . . , a
n
ucln(0, a
1
, . . . , a
n
) = ucln(a
1
, . . . , a
n
).
ucln(1, a
1
, . . . , a
n
) = 1 = ucln(−1, a
1
, . . . , a
n
).
ucln(a
1
, . . . , a
n
) = ucln(ucln(a

1
, . . . , a
n−1
), a
n
).
d = ucln(a
1
, . . . , a
n
) d > 0, d
uc(a
1
, . . . , a
n
), d d

∈ uc(1, a
1
, . . . , a
n
).
ucln(ka
1
, . . . , ka
n
) = |k|ucln(, a
1
, ucln(a
2

, . . . , a
n
)), k ∈ Z \ {0}.
d = ucln(a
1
, . . . , a
n
) d > 0, d
uc(a
1
, . . . , a
n
), x
1
, . . . , x
n
∈ Z d =

n
i=1
a
i
x
i
.
a, b, c, d ∈ Z \ {0}, a = bc + d ucln(a, b) = ucln(b, d).
ucln(0, a) = |a| a ∈ Z, a = 0.
a
1
, . . . , a

n
∈ Z
ucln(a
1
, . . . , a
n
)
I = {y =

n
j=1
a
j
x
j
| ∀x
j
∈ Z}. a
i
= 1.a
i
+

n
i=j=1
a
j
0 a
i
I. I = {0}. y ∈ I

−y ∈ I. I. d
I. d = ucln(a
1
, . . . , a
n
).
d ∈ uc(a
1
, . . . , a
n
) : a
i
= q
i
d + r
i
, 0 ≤ r
i
< d
d ∈ I, d =

n
j=1
a
j
x
j
, x
j
∈ Z.

r
i
= a
i
− q
i
d = a
i
− q
i
n

j=1
a
j
x
j
= a
1
(−q
i
x
1
) + ··· + a
i
(1 −q
i
x
i
) + ··· + a

n
(−q
i
x
n
) ∈ I.
r
i
∈ I i = 1, . . . , n. d
I r
i
∈ I, 0 ≤ r
i
< d, r
1
= ··· = r
n
= 0.
c ∈ uc(a
1
, . . . , a
n
) c|d. c ∈
uc(a
1
, . . . , a
n
) b
j
∈ Z a

j
= b
j
c j = 1, . . . , n. d =

n
j=1
a
j
x
j
= c(

n
j=1
b
j
x
j
) c|d. d = ucln(a
1
, . . . , a
n
).
a
1
, . . . , a
n
∈ Z. d
a

i
x
j
∈ Z
d =

n
j=1
a
j
x
j
.
a
1
, . . . , a
n
∈ Z,
d = ucln(a
1
, . . . , a
n
)
a
i
.
a
1
, . . . , a
n

∈ Z,
d > 0 uc(a
1
, . . . , a
n
)
a
i
d = ucln(a
1
, . . . , a
n
).
a, b ∈ Z, b = 0. a b
a = q
0
b + r
1
0 ≤ r
1
< |b|. r
1
= 0 ucln(a, b) = b. r
1
= 0,
b = q
1
r
1
+ r

2
0 ≤ r
2
< r
1
. r
2
= 0
r
1
= ucln(b, r
1
) = ucln(a, b) r
2
= 0,
n r
n−2
= q
n−1
r
n−1
+ r
n
0 ≤ r
n
< r
n−1
,
|b| > r
1

> r
2
> ··· ≥ 0 r
i
∈ N.
m
r
m−2
= q
m−1
r
m−1
ucln(a, b) = r
m−1
.
n > 2
a = 196, b = 38. 196 = 38.5+6, 38 = 6.6+2, 6 =
3.2. ucln(196, 38) = 2.
x, y ∈ Z d =
ax + by, d = ucln(a, b),
d = ucln(a, b) = r
m−1
.
d = r
m−1
= r
m−3
− q
m−2
r

m−2
= r
m−3
− q
m−2
(r
m−4
− q
m−3
r
m−3
)
= (q
m−2
q
m−3
+ 1)r
m−3
− q
m−2
r
m−4
.
r
j
d = ax + by.
a
1
, . . . , a
n

∈ Z,
ucln(a
1
, . . . , a
n
) = 1.
ucln(a
i
, a
j
) = 1 i, j =
1, . . . , n, i = j.
a
1
, . . . , a
n
∈ Z,
x
j

n
j=1
a
j
x
j
=
1.
a, b ∈ Z
ucln(a, b) = 1 ∃ x, y ∈ Z ax + by = 1.

a, b ∈ Z
ucln(a, b) = d ∃ x, y ∈ Z ax + by = d.
ucln(a, b) = d ucln(
a
d
,
b
d
) = 1.
∃ x, y ∈ Z
a
d
x +
b
d
y = 1. d
ax + by = d.
a, b, c ∈ Z
ucln(a, c) = 1, ucln(b, c) = 1 ucln(ab, c) = 1.
∃ x, y, z, t ∈ Z ax+cy = 1, bz+ct =
1. 1 = (ax + cy)(bz + ct) = abxy + c(axt + byz).
ucln(ab, c) = 1.
a, b, c ∈ Z. ax + by = c
x, y Z
a, b, c ∈ Z,
ax + by = c d = ucln(a, b)
c.
x
0
, y

0
. c = ax
0
+
by
0
d c d. c d.
c = dp. d = ucln(a, b) x
1
, y
1
ax
1
+ by
1
= d
p ax
1
p + by
1
p = dp = c.
x
0
= px
1
, y
0
= py
1
.

a
1
, . . . , a
n
∈ Z \ {0}. m
a
1
, . . . , a
n
m
a
i
.
a
1
, . . . , a
n
∈ Z \ {0}. m
a
1
, . . . , a
n
m
a
i
m
bc(a
1
, . . . , a
n

)
bcnn(a
1
, . . . , a
n
) a
1
, . . . , a
n
. m
a
1
, . . . , a
n
−m
bcnn(a
1
, . . . , a
n
)
bc(a
1
, . . . , a
n
).
a, b ∈ Z \{0}, ucln(a, b). bcnn(a, b) = |ab|.
a, b > 0.
m =
ab
ucln(a, b)

. m m =
b
ucln(a, b)
a =
a
ucln(a, b)
b m a, b. c
a b, a|c, c = bh h ∈ Z
c
ucln(a, b)
=
hb
ucln(a, b)
.
a
ucln(a, b)
|
bh
ucln(a, b)
ucln(
a
ucln(a, b)
,
b
ucln(a, b)
) =
1,
a
ucln(a, b)
|h. p ∈ Z h =

ap
ucln(a, b)
.
c = bh = b
ap
ucln(a, b)
=
ab
ucln(a, b)
p = mp c m.
m a b
a
1
, . . . , a
n
∈ Z \{0}
bcnn(a
1
, . . . , a
n
) = bcnn(bcnn(a
1
, . . . , a
n−1
), a
n
).
n.
n = 1, bcnn(a
1

) = a
1
. n = 2, bcnn(a
1
, a
2
) =
|a
1
a
2
|
ucln(a
1
.a
2
)
n = 1, 2.
n a = bcnn(a
1
, . . . , a
n−1
). m = bcnn(a, a
n
).
m a a
n
, m
a
1

, . . . , a
n
. c a
i
c
a
1
, . . . , a
n−1
c a. c
m. m = bcnn(a
1
, . . . , a
n
)
a
1
, . . . , a
n
m, d
m = bcnn(a
1
, . . . , a
n
) ⇐⇒ ucln(
m
a
1
, . . . ,
m

a
n
) = 1.
bcnn(a
1
, . . . , a
n
)
d
= bcnn(
a
1
d
, . . . , ,
a
n
d
).
d bcnn(a
1
, . . . , a
n
) = bcnn(da
1
, . . . , da
n
).
p > 1
q > 1
p p

p = {p
1
, . . . , p
s
}. q =

s
i=1
p
i
+
1 > 1. q
1
q p
i
,
p
i
. p. p
F
F = ∅. F = ∅. m ∈ F m > 2, F
m F.
m ∈ F m q
1
, q
2
> 1
m = q
1
q

2
. q
1
, q
2
< m q
1
, q
2
/∈ F.
q
1
= t
1
t
2
. . . t
h
, q
2
= u
1
u
2
. . . u
k
, t
i
, u
j

m = q
1
q
2
= t
1
t
2
. . . t
h
u
1
u
2
. . . u
k
.
m ∈ F. F
q > 1
p
1
, . . . , p
s
α
1
, . . . , α
s
q = p
α
1

1
p
α
2
2
. . . p
α
s
s
q. a, b
a = p
α
1
1
p
α
2
2
. . . p
α
s
s
q
u
1
1
. . . q
u
r
r

, b = p
β
1
1
p
β
2
2
. . . p
β
s
s
t
v
1
1
. . . t
v
h
h
,
q
i
a,
t
j
b.
ucln(a, b) = p
min(α
1

,β
1
)
1
p
min(α
2
,β
2
)
2
. . . p
min(α
s
,β
s
)
s
,
bcnn(a, b) = p
max(α
1
,β
1
)
1
p
max(α
2
,β

2
)
2
. . . p
max(α
s
,β
s
)
s
q
u
1
1
. . . q
u
r
r
t
v
1
1
. . . t
v
h
h
.
n
n
10

+ n
5
+ 1
n n
4
+ (n + 1)
4
n
10
+ n
5
+ 1 = n
10
− n + n
5
+ n + 1
= n(n
3
− 1)(n
6
+ n
3
+ 1) + (n
2
+ n + 1)(n
3
− n
2
+ 1)
= (n

2
+ n + 1)[ ].
n = 1, 2, 3, 4 n
4
+ (n + 1)
4
n = 5
5
4
+ 6
4
= 1921 = 113.17.
x
1
, x
2
x
2
− 6x + 1 = 0. a
n
= x
n
1
+
x
n
2
, n ∈ N.
a
n

a
n
a
n+2
= 6a
n+1
− a
n
= 35a
n
− 6a
n−1
. a
0
=
2, a
1
= 6, a
2
= 34, a
n
a
n+2
= 6a
n+1
− a
n
= 35a
n
− 6a

n−1
a
n
≡ −a
n−3
(mod 5).
a
0
= 2, a
1
= 6, a
2
= 34 a
n
a
0
, a
1
a
n
a
n+2
= 6a
n+1
− a
n
a
m
a
n+2

−a
n
a
0
, a
1
, a
2
a
n
a
n
k > 1. n
n = a
0
k
m
+ a
1
k
m−1
+ ···+ a
m−1
k + a
m
,
a
0
, . . . , a
m

∈ {0, 1, . . . , k −1}.
k > 1. n
n = a
0
k
m
+ a
1
k
m−1
+ ···+ a
m−1
k + a
m
,
a
0
, . . . , a
m
∈ {0, 1, . . . , k −1}.
n
0 ≤ n ≤ k − 1, n = n. n ≥ k m
k
m
≤ n < k
m+1
. a
0
∈ {1, . . . , k − 1}
a

0
k
m
≤ n < (a
0
+ 1)k
m
. 0 ≤ n
1
= n − a
0
k
m
< k
m
.
a
i
, m−i a
i
k
m−i
≤ n
1
< k
m−i+1
.
a
i
∈ {0, 1, . . . , k −1} n = a

0
k
m
+a
1
k
m−1
+···+a
m−1
k + a
m
.
n k

n = a
0
k
m
+ a
1
k
m−1
+ ···+ a
m−1
k + a
m
,
n = b
0
k

s
+ b
1
k
s−1
+ ···+ b
s−1
k + b
s
.
(a
0
k
m
+ ··· + a
m−1
k) − (b
0
k
s
+ ··· + b
s−1
k) + (a
m
−b
s
) = 0.
k a
m
−b

s
k. a
m
= b
s
.
m = s, a
i
= b
i
, ∀i,
n = a
0
k
m
+ a
1
k
m−1
+ ···+ a
m−1
k + a
m
n = (a
0
a
1
···a
m−1
a

m
)
k
.
k a
0
, a
1
, . . . , a
m
n k.
160 = 3.7
2
+ 1.7
1
+ 6 = (316)
7
,
95 = 1.3
4
+ 0.3
3
+ 1.3
2
+ 1.3
1
+ 2 = (10112)
3
.
a, b ∈ Z, b > 0. a

b
a = q
0
b + r
1
0 < r
1
< b,
b = q
1
r
1
+ r
2
0 < r
2
< r
1
,
r
1
= q
2
r
2
+ r
3
0 < r
3
< r

2
,
,
r
n−2
= q
n−1
r
n−1
+ r
n
0 < r
n
< r
n−1
,
r
n−1
= q
n
r
n
.
ucln(a, b) = r
n
.
a
b
a
b

= q
0
+
1
b
r
1
= q
0
+
1
q
1
+
1
r
1
r
2
= q
0
+
1
q
1
+
1
q
2
+

1
r
2
r
3
=
= q
0
+
1
q
1
+
1
q
2
+
1
q
3
+
1

+
1
q
n
.•
q
0

+
1
q
1
+
1
q
2
+
1
q
3
+
1

+
1
q
n
, (1),
q
0
q
1
, . . . , q
n
q
n
> 1. n
[q

0
; q
1
, . . . , q
n
].
a
b
=
[q
0
; q
1
, . . . , q
n
].
143
27
= [5; 3, 2, 1, 2],
−231
12
= [−20; 1, 3].
[q
0
; q
1
, . . . , q
n
]
a

b
.
[q
0
; q
1
, . . . , q
n
] =
a
b
= [p
0
; p
1
, . . . , p
m
].
m = n q
i
= p
i
i = 0, . . . , n,
n. n = 0, q
0
= [p
0
; p
1
, . . . , p

m
]. p
0
q
0
,
m = 0, q
0
= p
0
. n > 0,
n. [q
0
; q
1
, . . . , q
n
] = [p
0
; p
1
, . . . , p
m
],
q
0
= p
0
,
[0; q

1
, . . . , q
n
] =
a
b
−q
0
= [0; p
1
, . . . , p
m
]. [q
1
; . . . , q
n
] = [p
1
; . . . , p
m
].
n − 1 = m − 1 q
i
= p
i
i =
1, . . . , n.
[q
0
; q

1
, . . . , q
n
], (2).
g
0
= q
0
= [q
0
], g
1
= q
0
+
1
q
1
= [q
0
; q
1
], g
2
= q
0
+
1
q
1

+
1
q
2
= [q
0
; q
1
, q
2
], ,
{P
i
}, {Q
i
}, i = 0, 1, . . . , n,

P
0
= q
0
,
Q
0
= 1,

P
1
= q
1

q
0
+ 1,
Q
1
= q
1
,

P
m
= q
m
P
m−1
+ P
m−2
,
Q
m
= q
m
Q
m−1
+ Q
m−2
, m ≥ 2.
g
j
=

P
j
Q
j
j = 0, . . . , n.
j. j =
0 g
0
= q
0
=
P
0
Q
0
. j = m
g
m
=
P
m
Q
m
=
q
m
P
m−1
+ P
m−2

q
m
Q
m−1
+ Q
m−2
. g
m
= [q
0
; q
1
, . . . , q
m
], g
m+1
=
[q
0
; q
1
, . . . , q
m
, q
m+1
]. P
j
, Q
j
q

m
j < m,
g
m+1
g
m
q
m
q
m
+
1
q
m+1
,
g
m+1
=
(q
m
+
1
q
m+1
)P
m−1
+ P
m−2
(q
m

+
1
q
m+1
)Q
m−1
+ Q
m−2
=
(q
m
q
m+1
+ 1)P
m−1
+ q
m+1
P
m−2
(q
m
q
m+1
+ 1)Q
m−1
+ q
m+1
Q
m−2
.

g
m+1
=
q
m+1
P
m
+ P
m−1
q
m+1
Q
m
+ Q
m−1
=
P
m+1
Q
m+1
P
m−1
Q
m
− Q
m−1
P
m
= (−1)
m

1 ≤ m ≤ n;
P
m
, Q
m
g
m
P
m
Q
m
−
P
m−1
Q
m−1
=
(−1)
m+1
Q
m−1
Q
m
.
δ
m
= P
m−1
Q
m

− Q
m−1
P
m
. δ
m
= (−1)
m
.
m = 1 δ
1
= P
0
Q
1
− Q
0
P
1
= q
0
q
1
− q
0
q
1
− 1 = (−1)
1
.

m = 1. m > 1,
δ
m
= P
m−1
Q
m
− Q
m−1
P
m
= P
m−1
(q
m
Q
m−1
+ Q
m−2
) −Q
m−1
(q
m
P
m−1
+ P
m−2
)
= P
m−1

Q
m−2
− Q
m−1
P
m−2
= −δ
m−1
.
δ
m
= (−1)
m
.
P
j
, Q
j
P
m−1
Q
m
−Q
m−1
P
m
= (−1)
m
P
m

, Q
m
g
m
a
b
= [q
0
; q
1
, . . . , q
n
] > 0
b
a
= [0; q
0
, q
1
, . . . , q
n
].
a, b, c ∈ Z a, b
x, y ax + by = c, (3).
a, b, c ∈ Z a, b c
ucln(a, b), ax + by = c x
0
, y
0
∈ Z

ucln(a, b) = d. c d;
c d
d = 1, b = 0
a
b
a
b
= g
n
=
P
n
Q
n
.
a = P
n
, b = Q
n
.
a(−1)Q
n−1
+ bP
n−1
= (−1)
n
a(−1)
n+1
cQ
n−1

+ b(−1)
n
cP
n−1
= c.
x
0
= (−1)
n+1
cQ
n−1
, y
0
= (−1)
n
cP
n−1
∈ Z
x, y ∈ Z

ax + by = c,
ax
0
+ by
0
= c.
a(x−x
0
) = b(y
0

−y). a, b
+1, −1, x −x
0
b. x −x
0
= bt, t ∈ Z.
y − y
0
= −at. x = x
0
+ bt, y = y
0
− at, t ∈ Z.
143x −27y = 13.
−143
27
= [−6; 1, 2, 2, 1, 2]. P
i
, Q
i
P
0
= −6, Q
0
= 1; P
1
= −5, Q
1
= 1; P
2

= 2(−5) −6 = −16, Q
2
=
2.1+1 = 3; P
3
= −37, Q
3
= 7; P
4
= −53, Q
4
= 10; P
5
= −143, Q
5
= 27.
x
0
= (−1)
6
13.10 = 130, y
0
= (−1)
5
13.(−53) = 689
x = 130 −27t, y = 689 − 143t, t ∈
Z.
b, a
1
, a

2
, . . . , a
n
∈ Z a
i
a
1
x
1
+ a
2
x
2
+ ···+ a
n
x
n
= b, (4).
(4) (z
i
) b
d = ucln(a
1
, . . . , a
n
). (z
i
)
|z
i

| ≤ |b| + (n −1) max{|a
j
|}, i = 1, . . . , n.
x
1
, . . . , x
n
x
i
= q
i
a
n
+ z
i
, 0 ≤ z
i
< |a
n
|, i = 1, . . . , n −1,
q
i
, z
i
z
n
= x
n
+


n−1
i=1
a
i
q
i
.
b =
n−1

i=1
a
i
x
i
+ a
n
x
n
=
n−1

i=1
a
i
(q
i
a
n
+ z

i
) + a
n
(z
n
−
n−1

i=1
a
i
q
i
) =
n

i=1
a
i
z
i
.
(z
i
) |z
i
| < |a
n
|
i = 1, . . . , n − 1; z

n
|a
n
z
n
| = |b −
n−1

i=1
a
i
z
i
| ≤ |b| + (n −1)|a
n
|max{|a
j
|}.
|a
n
| |z
i
| ≤ |b| + (n − 1) max{|a
j
|}, i =
1, . . . , n.
{q
i
} q
i

> 0 i = 1, . . . , n.
a
0
, a
1
, a
2
, a
3
, . . . b
0
, b
1
, b
2
, b
3
, . . . .
N
0
= a
0
, N
1
= a
0
+
b
0
a

1
, N
2
= a
0
+
b
0
a
1
+
b
1
a
2
, ,
{a
i
} {b
i
}.
P
0
, P
1
, P
2
, P
3
, . . . Q

0
, Q
1
, Q
2
, Q
3
, . . .
P
0
= a
0
, P
1
= a
1
a
0
+ b
0
, Q
0
= 1, Q
1
= a
1
,

P
n+1

= a
n+1
P
n
+ b
n
P
n−1
,
Q
n+1
= a
n+1
Q
n
+ b
n
Q
n−1
, n ≥ 1.
N
n
=
P
n
Q
n
n ≥ 0.
a
0

+
b
0
a
1
+
b
1
a
2
+
b
2
a
3
+
b
3

+
b
n−1
a
n
=
P
n
Q
n
, ∀n ≥ 0.

n.
n = 0, 1 n.
a
0
+
b
0
a
1
+
b
1
a
2
+
b
2
a
3
+
b
3

a
n−1
+
b
n−1
a
n

=
P
n
Q
n
= (4).
(5) = a
0
+
b
0
a
1
+
b
1
a
2
+
b
2
a
3
+
b
3

a
n−1
+

b
n−1
a
n
+
b
n
a
n+1
a
n
a
n
+
b
n
a
n+1
. P
n
, Q
n
b
n
, a
n+1
.
P
n
Q

n
=
a
n
P
n−1
+ b
n−1
P
n−2
a
n
Q
n−1
+ b
n−1
Q
n−2
(5) =
(a
n
+
b
n
a
n+1
)P
n−1
+ b
n−1

P
n−2
(a
n
+
b
n
a
n+1
)Q
n−1
+ b
n−1
Q
n−2
=
(a
n
a
n+1
+ b
n
)P
n−1
+ a
n+1
b
n−1
P
n−2

(a
n
a
n+1
+ b
n
)Q
n−1
+ a
n+1
b
n−1
Q
n−2
=
a
n+1
(a
n
P
n−1
+ b
n−1
P
n−2
) + b
n
P
n−1
a

n+1
(a
n
Q
n−1
+ b
n−1
Q
n−2
) + b
n
Q
n−1
=
a
n+1
P
n
+ b
n
P
n−1
a
n+1
Q
n
+ b
n
Q
n−1

=
P
n+1
Q
n+1
.
n.
b
i
= 1, a
i
= q
i
, i = 0, 1, . . . , n, a
i
= b
i
= 0, ∀i > n,
n

i=1
1
i
=
1
1 −
1
2
3 −
2

2
5 −
3
2
7 −
4
2

−
(n −1)
2
2n −1
, ∀n ≥ 1.
b
0
= 1, a
n
= 2n −1, b
n
= n
2
Q
n
= n!, P
n
= n!

n
i=1
1

i
.
P
n
Q
n+1
− Q
n
P
n+1
= (−1)
n+1
b
0
. . . b
n
0 ≤ n;
P
n+1
Q
n+1
−
P
n
Q
n
=
(−1)
n
b

0
. . . b
n
Q
n+1
Q
n
;
P
n+1
Q
n+1
= a
0
+

n
i=0
(−1)
i
b
0
. . . b
i
Q
i+1
Q
i
.
lim

n→∞
N
n
= lim
n→∞
P
n
Q
n
.
δ
n
= P
n
Q
n+1
− Q
n
P
n+1
.
δ
n
= (−1)
n+1
n

i=0
b
i

.
n = 0 δ
0
= P
0
Q
1
− Q
0
P
1
= a
0
a
1
− a
0
a
1
− b
0
= −b
0
.
n = 0. n > 0,
δ
n
= P
n
Q

n+1
− Q
n
P
n+1
= P
n
(a
n+1
Q
n
+ b
n
Q
n−1
) −Q
n
(a
n+1
P
n
+ b
n
P
n−1
)
= b
n
(P
n

Q
n−2
− Q
n−1
P
n
) = −b
n
δ
n−1
.
δ
n
= (−1)
n+1
b
0
. . . b
n
.
P
n+1
Q
n+1
=

n
i=0
[
P

i+1
Q
i+1
−
P
i
Q
i
] + a
0
= a
0
+

n
i=0
(−1)
i
b
0
. . . b
i
Q
i+1
Q
i
.
lim
n→∞
N

n
= lim
n→∞
P
n
Q
n
P
n+1
+ P
n
= b
0
+ a
0
+

n
i=0
a
i+1
P
i
.
b
i
= 1, ∀i ≥ 1. P
i+1
= a
i+1

P
i
+ P
i−1
i = 1, 2, . . . , n, P
n+1
+ P
n
= b
0
+ a
0
+

n
i=0
a
i+1
P
i
.
P
0
, P
1
, P
2
, P
3
, . . .

P
0
= 0, P
1
= 1, P
n+1
= (n+1)P
n
+P
n−1
, n ≥ 1.
P
n+1
+ P
n
= 1 +

n
i=0
(i + 1)P
i
.
P
0
= 0!, P
1
= 0!1!, P
n+1
= (n + 1)!P
n

+
P
n−1
, n ≥ 1. P
n+1
+ P
n
= 1 +

n
i=0
(i + 1)!P
i
.
P
n+1
+P
n
= b
0
+a
0
+

n
i=0
a
i+1
P
i

a
i
= i, b
0
= 1,
P
n+1
+ P
n
= 1 +

n
i=0
(i + 1)P
i
.
P
n+1
+ P
n
= b
0
+ a
0
+

n
i=0
a
i+1

P
i
a
i
= i!, b
0
= 0,
P
n+1
+ P
n
= 1 +

n
i=0
(i + 1)!P
i
.
α /∈ Q. α = q
0
+
1
α
1
, q
0
= [α] ∈ Z, α
1
> 1. α /∈ Q
α

1
α
1
= q
1
+
1
α
2
, q
1
= [α
1
] ∈ N
+
, α
2
> 1.
n + 1 α
n
= q
n
+
1
α
n+1
, q
n
= [α
n

] ∈
N
+
, α
n+1
> 1. α /∈ Q α
n+1
α = q
0
+
1
q
1
+
1
q
2
+
1

, π
n
= q
0
+
1
q
1
+
1

q
2
+
1
q
3
+
1

q
n−1
+
1
q
n
n = 0, 1, . . . . α
π
i
α. α = [q
0
; q
1
, q
2
, . . .].
{P
i
}, {Q
i
}, i = 0, 1, . . . ,


P
0
= q
0
,
Q
0
= 1,

P
1
= q
1
q
0
+ 1,
Q
1
= q
1
,

P
n
= q
n
P
n−1
+ P

n−2
,
Q
n
= q
n
Q
n−1
+ Q
n−2
, n ≥ 2.
π
i
=
P
i
Q
i
i = 0, 1, . . . ;
P
n−1
Q
n
− Q
n−1
P
n
= (−1)
n
1 ≤ n;