n
n = ab 1 < a < b < n n
"
n k1 := n x
x x
2
m1 " N k1 - n = m2
1
2
n = k1 -m2 = (k1+m1)(k1-m1).
1
k2 := k1 + 1
m2 i
2
mi, ki ki - n = m2 ki < n mi
i
n = (ki + mi)(ki - mi) n
n
n n = ab
1 < a d" b < n n
a = b n a, b

2|(a Ä… b) n = (a+b)2 - (a-b)2
2 2
a+b a-b
k := , m :=
2 2
" "
k2 > n k > n k e" n
"
a+b
k = < b < n n
2
n = 27 308 533
2
k1 = H" 5 225, 7 = 5 226 k1 - n = 2 543 H" (50, 4)2
2
k2 = 5227 k2 - n = 12 966 = (114)2 m2 = 114
27 308 533 = (5 227 + 114)(5 227 - 114) = 5 341 · 5 113
 n " N
n Zn
n " N a1, ..., an " Z \ {0}, b " Z
a1x1 + a2x2 + · · · + anxn = b
x1, ..., xn
a1, ..., an,
x + 3y = 2, x1 + 2x2 - 7x3 = 0
ax + by = c x, y
NW D(a, b) | c
(=Ò!) ax + by = c
x0, y0 NW D(a, b) = d d|c
u, v a = ud, b = vd
c = ax0 + by0 c = d(ux0 + by0) d|c.
(Ð!=) NW D(a, b) = d d|c
ax + by = c c = kd
k d = au + bv u, v
k
c = a(ku) + b(kv),
ku, kv
(1) ax + by = c
a, b
(2)
ax+by = c
a, b
x0, y0
ax+by = c
x = x0 + bt, y = y0 - at, t " Z.
 x, y ax+by = c
ax0 + by0 = c = ax + by
a(x - x0) = b(y0 - y).
b a(x - x0) NW D(a, b) = 1
b|(x - x0)
a|(y0 - y) t, q
(x - x0) = bt, (y0 - y) = aq.
abt = baq t = q
x, y
t
x = x0 + bt, y = y0 - at.
4x + 22y = 3
NW D(4, 22) 3
3x - 2y = 1
NW D(3, -2) = 1
x0 = 1, y0 = 1
x = 1 + (-2)t = 1 - 2t, y = 1 - 3t, t " Z.
22x + 6y = 8 Z
NW D(22, 6)|8
NW D(22, 6)
11x + 3y = 4
11 = 3 · 3 + 2, 3 = 2 + 1 1 = 3 - 2 = 3 - (11 - 3 · 3) = 4 · 3 - 1 · 11,
1 = 4 · 3 - 1 · 11
4 = 16·3-4·11 x0 = -4, y0 = 16
x = -4 + 3t, y = 16 - 11t, t " Z.
·2
11 = 3 · 3 + 2 2 = 11 - 3 · 3 4 = 2 · 11 - 6 · 3 x0 = 2, y0 = -6.
NW D(a1, a2, a3) = NW D(NW D(a1, a2), a3)
NW D
a1x1 + a2x2 + · · · + anxn = b
x1, ..., xn NW D(a1, ..., an)|b.
6x+8y-3z = 2
NW D(6, 8, -3) = 1
x, y
6x + 8y = 2k 3x + 4y = k
2k - 3z = 2 2k - 3z = 2
k0 = 1, z0 = 0 k = 1 - 3t, z = -2t, t " Z.
x, y k
1 = 4 - 3 1 - 3t = 4(1 - 3t) - 3(1 - 3t) x0 = 3t - 1, y0 = 1 - 3t
x = 3t - 1 + 4m, y = 1 - 3t - 3m, m " Z.
x = -1 + 3t + 4m, y = 1 - 3t - 3m, z = -2t, t, m " Z
x - 3y + 2z - 4t = 1
x - 3y = a, z - 2t = b, a + 2b = 1