�

– Fermat –

�� � � �� � � � � �(

)

���� ��� ��

– Fermat

� ��� � ��� �

– – p. 1

Pell II (1/30)Pell .

• F0(X0, Y0) = X20 − NY 2

0

� �

.

•

�

F0(X0, Y0) Xi, Yi

�

Fi(Xi, Yi) (i = 1, 2, 3, . . .) .

• Fi(Xi, Yi) = (−1)i Xi = 1, Yi = 0

� �

,

�

Xi, Yi

.

���� ��� ��

– Fermat

� ��� � ��� �

– – p. 2

Pell II (2/30)Fi(Xi, Yi) Fi+1(Xi+1, Yi+1)�

.• Fi(Xi, Yi) = AX2

i −BXiYi −CY 2i ,

At2 − Bt − C = 0 ,mi

�.

• Fi+1(Xi+1, Yi+1) = −Fi(miXi+1 + Yi+1, Xi+1)�

.

���� ��� ��

– Fermat

� ��� � ��� �

– – p. 3

Pell II (3/30)F0(X0, Y0) = X2

0 − 23Y 20�

. F1(X1, Y1),

• t2 − 23 = 0 .• .

.

�m0 = 4

�

.

���� ��� ��

– Fermat

� ��� � ��� �

– – p. 4

Pell II (4/30)F1(X1, Y1)

�

�

,

F1(X1, Y1) = −F0(4X1 + Y1, X1)

= −(4X1 + Y1)2+23X2

1

= 7X2

1 − 8X1Y1 − Y 2

1

� �

(X1, Y1) = (1, 0) F1(X1, Y1) = (−1)1

, �

� �

.

���� ��� ��

– Fermat

� ��� � ��� �

– – p. 5

Pell II (5/30)F2(X2, Y2) ,7t2 − 8t − 1 = 0 ,

�

�

, m1 = 1

�

. �

F2(X2, Y2) = −F1(1 · X2 + Y2, X2)

= −7(X2 + Y2)2+8(X2 + Y2)X2+X2

2

= 2X22 − 6X2Y2 − 7Y 2

2

(X2, Y2) = (1, 0) F2(X2, Y2) = (−1)2

, �

� �

.

���� ��� ��

– Fermat

� ��� � ��� �

– – p. 6

Pell II (6/30)F3(X3, Y3) ,2t2 − 6t − 7 = 0 ,

�

�

, m2 = 3

�

. �

F3(X3, Y3) = −F2(3 · X3 + Y3, X3)

= −2(3X3 + Y3)2+6(3X3 + Y3)X3+7X2

3

= 7X23 − 6X3Y3 − 2Y 2

3

(X3, Y3) = (1, 0) F3(X3, Y3) = (−1)3

, �

� �

.

���� ��� ��

– Fermat

� ��� � ��� �

– – p. 7

Pell II (7/30)Fi(Xi, Yi) i = 4

� �

.

F0(X0, Y0) = X2

0 − 23Y 2

0 , m0 = 4

F1(X1, Y1) = 7X2

1 − 8X1Y1 − Y 2

1 , m1 = 1

F2(X2, Y2) = 2X22 − 6X2Y2 − 7Y 2

2 , m2 = 3

F3(X3, Y3) = 7X23 − 6X3Y3 − 2Y 2

3 , m3 = 1

F4(X4, Y4) = X2

4 − 8X4Y4 − 7Y 2

4 , m4 = 8

���� ��� ��

– Fermat

� ��� � ��� �

– – p. 8

Pell II (8/30)Fi(Xi, Yi) i = 9

�

�

.

F4(X4, Y4) = X2

4 − 8X4Y4 − 7Y 2

4 , m4 = 8

F5(X5, Y5) = 7X2

5 − 8X5Y5 − Y 2

5 , m5 = 1

F6(X6, Y6) = 2X26 − 6X6Y6 − 7Y 2

6 , m6 = 3

F7(X7, Y7) = 7X27 − 6X7Y7 − 2Y 2

7 , m7 = 1

F8(X8, Y8) = X2

8 − 8X8Y8 − 7Y 2

8 , m8 = 8

F9(X9, Y9) = 7X2

9 − 8X9Y9 − Y 2

9 , m9 = 1

���� ��� ��

– Fermat

� ��� � ��� �

– – p. 9

Pell II (9/30)

• Fi(Xi, Yi) = (−1)i Xi = 1, Yi = 0

� �

,

�

Xi, Yi

.

, Fi(Xi, Yi) = (−1)i

�

, (Xi, Yi) = (1, 0)

�

i

i = 4, 8, 12, . . .

�

.

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 10

Pell II (10/30)

� (X4, Y4) = (1, 0)

�

,

Xi = miXi+1 + Yi+1, Yi = Xi+1

�

,

(X3, Y3) = (X4 + Y4, X4) = (1, 1)

(X2, Y2) = (3X3 + Y3, X3) = (4, 1)

(X1, Y1) = (X2 + Y2, X2) = (5, 4)

(X0, Y0) = (4X1 + Y1, X1) = (24, 5)

� �

, (24, 5)

�

.

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 11

Pell II (11/30)

�

,,

.1 x2 − 35y2 = 1

.. (x, y) = (6, 1).

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 12

Pell II (12/30)Fi(Xi, Yi)

� � �.

F0(X0, Y0) = X2

0 − 23Y 2

0 , m0 = 4

F1(X1, Y1) = 7X2

1 − 8X1Y1 − Y 2

1 , m1 = 1

F2(X2, Y2) = 2X22 − 6X2Y2 − 7Y 2

2 , m2 = 3

F3(X3, Y3) = 7X23 − 6X3Y3 − 2Y 2

3 , m3 = 1

F4(X4, Y4) = X2

4 − 8X4Y4 − 7Y 2

4 , m4 = 8

F5(X5, Y5) = 7X2

1 − 8X1Y1 − Y 2

1 , m5 = 1

F6(X6, Y6) = 2X2

6 − 6X6Y6 − 7Y 2

6 , m6 = 3

F7(X7, Y7) = 7X27 − 6X7Y7 − 2Y 2

7 , m7 = 1

F8(X8, Y8) = X28 − 8X8Y8 − 7Y 2

8 , m8 = 8

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 13

Pell II (13/30)√

23

�

,�

. (

�

,

�.)

√23 = 4 +

1

1+

1

3+

1

1+

1

8+

1

1+· · ·

�

,

� � �

.

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 14

Pell II (14/30)Fi(Xi, Yi)

�

mi,� � �

.• Fi(Xi, Yi) 4 .

• mi

�

.

� �

Fi(Xi, Yi),mi� �

.

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 15

Pell II (15/30)

� �

.

Fi(Xi, Yi) = AX2

i − BXiYi − CY 2

i

,

Gi(Xi, Yi) = CX2i − BXiYi − AX2

i

�

. ,

F1(X1, Y1) = 7X21 − 8X1Y1 − 1Y 2

1

G1(X1, Y1) = 1X21 − 8X1Y1 − 7Y 2

1

F2(X2, Y2) = 2X2

2 − 6X2Y2 − 7Y 2

2

G2(X2, Y2) = 7X2

2 − 6X2Y2 − 2Y 2

2� �� � �� ��

– Fermat

� �� � � �� �

– – p. 16

Pell II (16/30)

� �

Gi(Xi, Yi) = CX2i − BXiYi − AY 2

i ,, Ct2 − Bt − A = 0 ,

ni�

.

�

�

, .

Gi−1(Xi−1, Yi−1) = −Gi(niXi−1 + Yi−1, Xi−1)

�

�

. ,

G1(X1, Y1) = X21 − 8X1Y1 − 7Y 2

1 , n1 = 8

G2(X2, Y2) = 7X2

2 − 6X2Y2 − 2Y 2

2 , n2 = 1

G3(X3, Y3) = 2X2

3 − 6X3Y3 − 7Y 2

3 , n3 = 3

G4(X4, Y4) = 7X2

4 − 8X4Y4 − Y 2

4 , n4 = 1

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 17

Pell II (17/30)

−G4(n4X3 + Y3, X3) = −7(X3 + Y3)2

+8(X3 + Y3)X3 + X2

3

= 2X2

3 − 6X3Y3 − 7Y 2

3

= G3(X3, Y3)

−G3(n3X2 + Y2, X2) = −2(3X2 + Y2)2

+6(3X2 + Y2)Y2 + 7X2

2

= 7X2

2 − 6X2Y2 − 2Y 2

2

= G2(X2, Y2)

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 18

Pell II (18/30)Fi(Xi, Yi)

�

Gi(Xi, Yi)

� �

�

.

Fi = AX2

i − BXiYi − CY 2

i

mi = At2 − Bt − C = 0

Gi = CX2

i − BXiYi − AY 2

i

ni = Ct2 − Bt − A = 0

Fi+1 = Fi(miXi+1 + Yi+1, Xi+1)

Gi−1 = Gi(niXi−1 + Yi−1, Xi−1)

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 19

Pell II (19/30)1 N

�

. Pellx2 −Ny2 = 1 , Fi(Xi, Yi)�

, Fi(Xi, Yi) p

. ,

F1 = Fp+1, F2 = F2+p . . .

.. .

Fi(Xi, Yi) = AiX2i − BiXiYi − CY 2

i

�

�

. i ≥ 1 Ai, Bi, Ci�

, |Ai − Ci| < Bi.B2

i + 4AiCi = 4N .

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 20

Pell II (20/30)i = 1 . [

√N ] = a

� �

A1 = N − n2, B1 = 2n,C1 = 1

�

.

i = k

�

.� �

, Fi+1

,

Ai+1 = −(Aim2i − Bmi − Ci), Bi+1 = 2Aimi − Bi

Ci+1 = Ai

�

.

�

B2i+1 + 4Ai+1Ci+1 = 4N

. mi Ait2 − Bit − C = 0

�

, Ai+1 > 0.

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 21

Pell II (21/30)

Bi+1 − (Ai+1 − Ci+1) = Ai(mi + 1)2

−Bi(mi + 1) − Ci

Bi+1 + (Ai+1 − Ci+1) = −Ai(mi − 1)2

+Bi(mi − 1) + Ci

�

. mi Ait2 − Bit − C = 0� � � � �

�

> 0�

, |Ai+1 − Ci+1| < Bi+1

Bi+1 > 0 ..

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 22

Pell II (22/30)Fi = AiX

2i −BiXiYi −CiY

2i

Ai, Bi, Ci .• Ai, Bi, Ci .

• B2i + 4AiCi = 4N

�

(Ai, Bi, Ci)

� � �

. �

i, p ,

Fi = Fi+p

�

.� �

Gi = Gi+p

. � �� � �� ��

– Fermat

� �� � � �� �

– – p. 23

Pell II (23/30)Gi Gi−1

� �

Gi+p Gi+p−1� � � �

,

Gi = Gi+p ⇒ Gi−1 = Gi+p−1 ⇒ Gi−2 = Gi+p−2

� � � �

,

G1 = Gp+1

.

�

F1 = Fp+1

�

,

�

� . 2

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 24

Pell II (24/30)1 Pell

Fi(Xi, Yi)

�

, i

Fi = (−1)i (1, 0)

�

. , Pell

�

.. p ,

F1 = Fp+1

�

.

� �

Fp�

Fp+1�

,

Ap = Cp+1

.

� �

,

Fp+1 = F1 = (N − m20)X

2 − 2m0XY − Y 2

�

, Ap = Cp+1 = 1. � �� � �� ��

– Fermat

� �� � � �� �

– – p. 25

Pell II (25/30)

�

Fp = X2 − BpXY − CpY2

�

, p Fp = (−1)p (1, 0)

�

. p , F2p

�

� �

F2p (1, 0)

�

. 2

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 26

Pell II (26/30)2 N

�

.

�

� √N .

. x2 − Ny2 ,

�Fi, mi .

�

Fi

�

ξi =Bi + 2

√N

2Ai

� �

.� �

.

√N = m0 +

1

ξ1

, ξi = mi +1

ξi+1

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 27

Pell II (27/30)A1 = N − m2

0, B1 = 2m0 ,

1

ξ1

=N − m2

0√N + m0

=√

N − m0

. Bi+1 = 2Aimi − Bi�

1

ξi − mi

=2Ai

Bi − 2Aimi + 2√

N

= − 2Ai

Bi+1 − 2√

N

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 28

Pell II (28/30)

� �

4N 2 = B2i + 4AiCi Ai+1 = Ci

�

1

ξi − mi

=2Ai(Bi+1 + 2

√N)

B2i+1

− 4N

=Bi+1 + 2

√N

2Ai+1

= ξi+1

�

.

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 29

Pell II (29/30)√

N = m0 +1

ξ1

, ξi = mi +1

ξi+1

√N

√N = m0 +

1

ξ1

= m0 +1

m1+

1

ξ2

= m0 +1

m1+

1

m2+

1

ξ3

= m0 +1

m1+

1

m2+

1

m3+

1

ξ4� �� � �� ��

– Fermat

� �� � � �� �

– – p. 30

Pell II (30/30)

� �

,

√N = m0 +

1

m1+

1

m2+

1

m3+

1

m4+· · ·

�

. pm1 = mp+1 ,

� � �

.

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 31

Fermat (1/2)

�

Fermat

� � �

� �

,• Fermat .• p = n2 + m2

�

.•

�

Pythagoras .• Pell .

�

,

� � �

,� � � �

.

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 32

Fermat (2/2)

� � � �

� �

,

� � �

� .Fermat

, Fermat .� � �

.• Fermat

� �

� �

� � � �

.•

� �

.• , � .• ,

�

� �

� � � �

� �

� .

� �� � �� ��

– Fermat

� �� � � �� �

– – p. 33

足立恒雄 (1994,2006)， フェルマーの大定理―整数論の源流，日本評論社／ちくま学芸文庫，より引用

とこ

ヨーロッパ

足立恒雄、三宅 克哉 (1987)， 数論―歴史からのアプローチ，日本評論社／ちくま学芸文庫にて復刊予定，より引用

Fermat (1/2)Fermat , Fermat

Huygens .

, ,,

Fermat. .

– Fermat – – p. 34

りとう

と とこっうう とこ

っェ ユ っ

Fermat (2/2),. ,

,

., ,

...( )...,

,,

,.

. ,. – Fermat – – p. 35

足立恒雄、三宅 克哉 (1987)， 数論―歴史からのアプローチ，日本評論社／ちくま学芸文庫にて復刊予定，より引用

こう と つここ

こ とこ とくう

と り とく こ く

くつ と く

っ こと と とこ

ょう とことっ

う とこ ょうう

こ

とっ ょう く り

足立恒雄 (1994,2006)， フェルマーの大定理―整数論の源流，日本評論社／ちくま学芸文庫，より引用

とこ

ヨーロッパ

こ

こ

く