�

– Fermat –

�� � � �� � � � � �(

)

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– Fermat

� ��� � ��� �

– – p. 1

( ) 3 .⇒ ( ) 3 = 3 × 1

�

.

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– Fermat

� ��� � ��� �

– – p. 2

Gauss1

�

a, b a + bi

�

Gauss

� �

.

� �

i

�

i2 = −1.1 Gauss• 2, 3, 4 .• 1 + i, 2 − 3i .

���� ��� ��

– Fermat

� ��� � ��� �

– – p. 3

Gauss2

�

Gauss α, β �

�

, βα

�

, Gauss γ�

,β = αγ

� � �

.2• 5 + 5i 1 + 2i .

5 + 5i = (1 + 2i)(3 − i).

• 5 1 − 2i .5 = (1 − 2i)(1 + 2i).

���� ��� ��

– Fermat

� ��� � ��� �

– – p. 4

Gauss3 Gauss ε

�

ε′ε = 1

�Gauss ε′

� �

.1 Gauss ±1,±i .. a + bi

� �

.�

Gaussc + di

�

, (a + bi)(c + di) = 1�

.(a − bi)(c − di) = 1

�,

,

(a2 + b2)(c2 + d2) = 1.

a, b, c, d , a = ±1, b = 0a = 0, b = ±1.

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– Fermat

� ��� � ��� �

– – p. 5

Gauss4 Gauss π

�

,Gauss α, β αβ π ,α β π

� �

, π

� �

.3• 2 . , 2 = (1 − i)(1 + i)

, 1 ± i 2 .• 5 . , 2 = (1− 2i)(1 + 2i)

, 1 ± 2i 5 .• 4m + 1 Gauss

�

�

,

� �.

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– Fermat

� ��� � ��� �

– – p. 6

�

1 Gauss α π1, . . . , πn

α = πm1

1· · ·πms

s

�

. α

α = ρn1

1· · · ρnt

t

�

� �

, s = t

�

,

�

πi = εiρi, 1 ≤ i ≤ t

�

.

� �

εi .

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– Fermat

� ��� � ��� �

– – p. 7

• 4

�

.

4 = 2 × 2 = −2i × 2i

• 5 + 5i

�

.

5 + 5i = (1 + 2i)(1 − 2i)(1 + i)

= (2 − i)(1 − 2i)(−1 + i)

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– Fermat

� ��� � ��� �

– – p. 8

Pythagoras (1/4)2 x, y, z x2 + y2 = z2 , x, y�

, u, v

x = u2 − v2, y = 2uv, z = u2 + v2

�

..

� �

Gauss �

. .

x2 + y2 = z2 ⇔ (x + iy)(x − iy) = z2

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– Fermat

� ��� � ��� �

– – p. 9

Pythagoras (2/4)1 x ± yi .

. x ± yi

�

π �

�

.

�

π 2x 2yi�

.

� �

π 2

� �

. z2 π

�

, z π

�

. � πz

�

2 .

� �

z 2

�

�

,

�

.

� π 2 .

�

π x, y�

, x, y

�

,

�

.

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 10

Pythagoras (3/4)x ± yi . �

(x + yi)(x − yi) = z2

x + yi

�

.

x + yi = ε(u + vi)2, ε = ±1,±i, u, v .

,

x = ±(u2 − v2), y = ±2uv, z = ±(u2 + v2)

.

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 11

Pythagoras (4/4), .

• �

� �

.• � � �

.

�

Fermat� �

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– Fermat

� �� � � �� �

– – p. 12

�

1 f(x), g(x), h(x) �

f(x), g(x)

�

,

f(x)2 − g(x)2 = h(x)2

�

.

� �

u(x), v(x) ,

f(x) = u(x)2 + v(x)2, g(x) = 2u(x)v(x),

h(x) = u(x)2 − v(x)2

� � �

.

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 13

(1/5)Fermat 1601 ,

• 1438 , .• 1453 ,

� � �

.

�

�

�

,

�

,

� �

.� �

(Euclid )� �

� .

� � � �

�

, ,.

� �

�

.

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– Fermat

� �� � � �� �

– – p. 14

(2/5)

x3 + ax2 + bx + c = 0

x = t − a/3� �

,

t3 + dx + e = 0

� � �.

� �

d = b −a2

3, e = c −

ab

3+

2a3

27

.

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– Fermat

� �� � � �� �

– – p. 15

(3/5)1 ω = −1 +

√3i/2

�

� �

.

ω2 + ω + 1 = 0

ω3 = 1

�

ω �

(t − u − v)(t − uω − vω2)(t − uω2 − vω)

�

.

�

,

t3 − 3uvt − u3 − v3

�

.

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 16

(4/5)

�

−3uv = d, −u3 − v3 = e

u, v ,

t = u + v, ωu + ω2v, ω2u + ωv

�

.

u3v3 = −d3

27, u3 + v3 = −e

� �

,�

,

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 17

(5/5)u3, v3

s2 + es −d3

27= 0

� �

,

�

u, v .� �� � �� ��

– Fermat

� �� � � �� �

– – p. 18

2

x3 − 6x2 + 11x − 6 = 0

. ( . ,

�

� � �

.)� �� � �� ��

– Fermat

� �� � � �� �

– – p. 19