1 1
6 12
1
7
d" 3 Z
x2 + y2 = z2 ax + by = c
xn + yn = zn
x, y, z n > 2
2
X2 + Y = Z2
x, y, z y
m, n
m > n
x = m2 - n2, y = 2mn, z = m2 + n2.
n " N
n Zn
xn + yn = zn.
 n e" 3
n = 1 x+y -z = 0
n = 2 x2 +y2 = z2
n " N, a1, ..., an, b " Z
a1x1 + a2x2 + · · · + anxn = b
x1, ..., xn
x + 3y = 2, x1 + 2x2 - 7x3 = 0
a, b, c " Z, a, b = 0 ax + by = c

x, y NW D(a, b)
c
ax + by = c
x0, y0 NW D(a, b) = d d|c
u, v " Z a = ud, b = vd
c = ax0 + by0 c = d(ux0 + by0) d|c.
NW D(a, b) = d d|c
ax + by = c k c = kd
u, v au + bv = d. k
a(ku) + b(kv) = c,
ku, kv
ax + by = c
a, b
ax + by = c
a, b
x0, y0 " Z
ax + by = c
b a
x = x0 + t, y = y0 - t, t " Z.
NW D(a, b) NW D(a, b)
 x0, y0
x, y ax0 + by0 = c =
ax + by
a(x - x0) = b(y0 - y).
d := NW D(a, b) r, s a = dr, b = ds
NW D(r, s) = 1 d
r(x - x0) = s(y0 - y).
s r(x - x0) NW D(r, s) = 1
s|(x - x0) r|(y0 - y)
t1, t2 (x - x0) = st1, (y0 - y) = rt2 st1 rt2
rst1 = srt2 t1 = t2 =: t.
t
b a
x = x0 + st = x0 + t, y = y0 - rt = y0 - t,
d d
3x - 2y = 1
NW D(3, -2) = 1 1
x0 = 1, y0 = 1
x = 1 + (-2)t = 1 - 2t, y = 1 - 3t, t " Z.
3(1 - 2t) - 2(1 - 3t) = 1
11x + 3y = 4
NW D(11, 3) = 1
11 = 3 · 3 + 2, 3 = 2 + 1 1 = 3 - 2 = 3 - (11 - 3 · 3) = 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
n
n " N, a1, a2, ..., an, b " Z, a2 > 0
i
i=1
a1x1 + a2x2 + · · · + anxn = b x1, ..., xn
NW D(a1, ..., an)|b
6x + 8y - 3z = 2
NW D(6, 8, -3) = 1
x, y
2k = 6x + 8y '" 2k - 3z = 2
k0 = 1, z0 = 0 k = 1 - 3t, z = -2t, t " Z
x, y k
2k = 6x + 8y 3x + 4y = k 3x + 4y = 1 - 3t.
1 = 4 - 3 1 - 3t = 4(1 - 3t) - 3(1 - 3t),
3x + 4y = -3(1 - 3t) + 4(1 - 3t) = 3(3t - 1) + 4(1 - 3t),
x0 = 3t - 1, y0 = 1 - 3t x = 3t - 1 + 4m, y = 1 - 3t - 3m, m " N
x = -1 + 3t + 4m, y = 1 - 3t - 3m, z = -2t, t, m " N
x-3y+2z-4t =
1 x-3y = a, z-2t = b, a+2b = 1