The Work of Sophie Germain and Niels Henrik Abel on Fermat’s Last Theorem

Lene-Lise Daniloff

Master’s Thesis, Spring 2017

This master’s thesis is submitted under the master’s programme Mathematics, with programme option Mathematics, at the Department of Mathematics, University of Oslo. The scope of the thesis is 60 credits.

The front page depicts a section of the root system of the exceptional Lie group E₈, projected into the plane. Lie groups were invented by the Norwegian mathematician Sophus Lie (1842–1899) to express symmetries in differential equations and today they play a central role in various parts of mathematics.

a, b, c n > 2

aⁿ = bⁿ + cⁿ aⁿ = bⁿ + cⁿ

n = 3 n = 4

nn = mp

(a^m)^p = (b^m)^p + (c^m)^p

p

aⁿ = bⁿ+cⁿ

• a, b, c
• a
• b

aⁿ = bⁿ + cⁿ

c

aⁿ = bⁿ + cⁿ

• n
• n > 2
• n

(−a)ⁿ = −aⁿ

• n
• a, b, c

na, b a, b

• c
• n

• c
• n

• a
• p
• a

a^p−1 ≡ 1 (mod p)

φ : Z₊ → Z₊

φ

φ(n)

• n
• n

• a
• n

a^φ(n) ≡ 1 (mod n)

p

φ(p) = p − 1

n

• φ
• φ

nn

• g
• g

k = φ(n) g^k ≡ 1 (mod n)

p

φ(p − 1)

pnmgpn

• n
• k

m ≡ g^k (mod n)

• p
• g
• p

g, g², ..., g^p−1

• n
• n = 2, n = 4, n = p^k

• k > 0
• n = 2p^k
• p > 2

ꢀ

• p
• g
• p

g^p−1 = 1

• n
• p

• n
• m
• n

m

a, n, m m

• a
• n

xⁿ ≡ a

(mod m)

• n = 3
• m = 5

• 1, 2, 3
• 3

x

x³

(mod 5)

• n = 3
• m = 7
• 3

x

x³

(mod 7)
7

• 3
• 7
• 3
• 5

• p
• n
• p

• u, v
• n

uⁿ−vⁿ

• φ(u, v) =
• uⁿ − vⁿ = (u − v)φ(u, v)

u−v

φ(u, v)
(u − v)(uⁿ⁻¹ + uⁿ⁻²v + ... + uvⁿ⁻¹ + vⁿ⁻¹

)

φ(u, v) = u − v

= uⁿ⁻¹ + uⁿ⁻²v + ... + uvⁿ⁻¹ + vⁿ⁻¹

• n
• xyz

np

xⁿ + yⁿ + zⁿ ≡ 0 (mod p) xⁿ ≡ n (mod p)

• x ≡ 0
• y ≡ 0

z ≡ 0 (mod p)

xn

• x, y, z
• n

zⁿ

xⁿ + yⁿ + zⁿ = 0

n

−xⁿ =

yⁿ +zⁿ = (y+z)(yⁿ⁻¹ −yⁿ⁻²z+...−yzⁿ⁻² +zⁿ⁻¹) = (y+z)φ(y, −z) q

• y ≡ −z (mod q)
• φ(y, −z) ≡ 0 (mod q)

nyⁿ⁻¹ ≡ 0

(mod q)

• q|n
• q|yⁿ⁻¹
• q|n
• q = n

n | xyz

• x
• y

y +z nn|y φ(y, −z)

• −yⁿ = (x + z)φ(x, −z)
• −zⁿ = (x + y)φ(x, −y)

a, b, c, α, β, γ

y + z = aⁿ x + y = bⁿ x + z = cⁿ φ(y, −z) = αⁿ φ(x, −y) = βⁿ φ(x, −z) = γⁿ

−x = aα −y = bβ −z = cγ p

2x = bⁿ + cⁿ − aⁿ

• x ≡ 0 (mod p)
• bⁿ + cⁿ + (−a)ⁿ ≡ 0 (mod p)

• a, b
• c
• p
• b
• c

• p
• x, y

x, z

a ≡ 0 (mod p)

y + z = aⁿ

y ≡ −z (mod p) αⁿ = φ(y, −z) ≡ nyⁿ⁻¹ (mod p) βⁿ = φ(x, −y) ≡ yⁿ⁻¹ (mod p)
⇒ αⁿ ≡ nβⁿ (mod p)

β ≡ 0 (mod p)

g

gβ ≡ 1 (mod p)

gⁿ

αⁿgⁿ ≡ n (mod p)

ꢀ

n < 100 p

5

• 11
• n = 5
• p = 11

• 5
• 1, 0

n = 5

• −1
• 11
• ±1 ± 1 ± 1 ≡ 0

• n
• p

n

• p
• p

• n
• p

n

pn

pn

x

• n = 5
• p = 11

• 5
• 11

x⁵

(mod 11)
5
11

n = 5

• p = 7
• 5
• 7

x

x⁵

(mod 7)
5

1, 2, 3, 4, 5, 6 p

• n
• p

• x, y
• z

xⁿ +yⁿ +zⁿ ≡ 0 (mod p)

• x ≡ 0 (mod p) y ≡ 0 (mod p)
• z ≡ 0 (mod p)

• p
• n

⇒

p

• n
• a

b

aⁿ ≡ bⁿ + 1 (mod p)

1 ≤ a, b ≤ p − 1

aⁿ − bⁿ − 1ⁿ ≡ 0 (mod p)

a

• b
• p
• a, b < p

⇒

• p
• n

xⁿ + yⁿ + yⁿ ≡ 0 (mod p)

x, y

ax ≡ y (mod p)

• z
• p
• a, b

bx ≡ z (mod p)

• p
• a

• b
• y, z

p

xⁿ + (ax)ⁿ + (bx)ⁿ ≡ 0 (mod p)
⇒ 1 + aⁿ + bⁿ ≡ 0 (mod p)

• p
• n

ꢀ

xⁿ + yⁿ + zⁿ = 0

n

• x, y
• z

• n
• x, y, z

• n
• p
• n

p = 2Nn + 1

N

• p
• n
• p
• n
• p − 1

pp

p = 2

p

• n
• p − 1

• m
• n = Km
• p − 1 = Mm

K, M p − 1 = ^M_K n = 2Nn p

MK

• 2N =
• p = 2Nn + 1

• n
• p

• p − 1
• n

a, b

a(p−1)+bn = 1

• u, v, w
• p

u + v + w ≡ 0 (mod p)

u^bn = u(u^{p−1 −a}

• )
• ≡ u (mod p)

• v^bn ≡ v (mod p)
• w^bn ≡ w (mod p)

(u^b)ⁿ + (v^b)ⁿ + (w^b)ⁿ ≡ 0 (mod p)

p

• p
• uvw

ꢀ

n

p = 2Nn + 1

• n
• N

• 2N
• n

p

xⁿ ≡ b (mod p) b^2N ≡ 1 (mod p)

0 < b < p

xⁿ ≡ b (mod p) b^2N ≡ (xⁿ)^2N ≡ x^p−1 ≡ 1 (mod p)

• b^2N ≡ 1 (mod p)
• b = 1

b = 1 x = 1

p

xⁿ ≡ b (mod p)

• p
• g

g^p−1 ≡ 1 (mod p)

p − 1

1 ≡ b^2N ≡ g^p−1 = g^2Nn = (gⁿ)^2N (mod p) b ≡ gⁿ (mod p)

• 2N
• b

b^2N ≡ 1 (mod p)

• g
• p

gⁿ, g²ⁿ, ..., g^2Nn

ꢀ

• k
• n
• p

p = 2Nn + 1

nn

• p − k
• n
• p

(p − k)ⁿ ≡ kⁿ (mod p)

n

k ≡ xⁿ (mod p) p − k ≡ −k ≡ −xⁿ = (−x)ⁿ (mod p)

p − k nn

ꢀ ꢁ

n

X

ni

• (p − k)ⁿ =
• (−1)ⁱ
• pⁿ⁻ⁱkⁱ ≡ kⁿ (mod p)

i=0

ꢀ

n = 3 n = 3

• N
• p > 13

p = 6N + 1

1, 2,
. . . , 6N

p − 1 = 6N

2N

2N − 1

4N

4N − 2(2N − 1) = 2

p > 13

2³ = 8

84

= 2

6N

2

1, 5, 8, 11..., 6N − 10, 6N − 7, 6N − 4, 6N
N ≥ 5

p ≥ 31
27 = 3³
27 ≡ 0 (mod 3)

6N

• p < 31
• p

4³ = 64 ≡ 7 (mod 19)

p

• p
• r, s

p

g^q = 2

• r − s = 2
• g

q

• 3 | q
• q = 3k + 1
• q = 3k − 1

kr + s

• r + s ≡ 0 (mod p)
• 2 = r − s ≡ 2r (mod p)

r < p r = 1

r ≡ 1 (mod p)

r − s = 2 m

r + s ≡ 0 (mod p) r + s ≡ g^m (mod p)

• 3|m
• r+s

• rs⁻¹ +1 ≡ g^ms⁻¹ (mod p)
• r+s ≡ g^m (mod p)

• m = 3l + 1
• m = 3l − 1

l ∈ N

m = 3l + 1 q = 3k − 1

• m = 3l±1
• m = 3l−1

q = 3k ∓ 1 q = 3k+1 r² − s² = (r − s)(r + s)
= 2(r + s) ≡ g^3k∓1g^3l±1 (mod p)

= g^3(k+l) r²s⁻² −1 ≡ g^3(k+l)s⁻² (mod p)

• m = 3l + 1
• q = 3k + 1

3l ± 1 m = 3l − 1

• q = 3k − 1
• 3k ± 1

2r = (r − s) + (r + s)
⇒ g^3k±1r ≡ g^3k±1 + g^3l±1 (mod p)
⇒ r ≡ 1 + g^3(k−l) (mod p)

ꢀ

n = 3 p = 6N + 1 n = 3

n = 3

n > 3

2Nn + 1

• n > 3
• n

p = 2Nn + 1

N = 1, 2, 4, 5
Nn

p = 2Nn + 1

• n
• N

p = 2Nn + 1

• n
• p

xⁿ + yⁿ = zⁿ

• x + y, z − y
• z − x

n²ⁿ⁻¹

n

• n = 5
• 11, 41, 71

x + y zⁿ = (x + y)φ(x, −y)

101

n

zⁿ

5⁹ · 11⁵ · 41⁵ · 71⁵ · 101⁵

xⁿ + yⁿ = zⁿ

x, y, z

• x, y
• z

x, y, z

n²ⁿ⁻¹

nn|z

• x+y
• n

• z − y
• z − x

• n
• x
• y

• x, y, z
• n

x + y = aⁿ z − x = bⁿ z − y = cⁿ φ(x, −y) = αⁿ φ(x, z) = βⁿ φ(y, z) = γⁿ aα = z bβ = y cγ = x

• n
• z

• n
• x + y ≡

xⁿ + yⁿ = zⁿ ≡ 0 (mod n)

s = x + y

xⁿ + yⁿ

x + y φ(x, −y) =

(s − y)ⁿ + yⁿ
=

s

ꢀ ꢁ

n−1

X

1

ni

• =
• (−1)ⁱ

sⁿ⁻ⁱyⁱ

s

i=0

• ꢀ ꢁ
• ꢀ
• ꢁ

• n
• n

= sⁿ⁻¹

−

sⁿ⁻²y + ... +

yⁿ⁻¹

1

n − 1

• φ(x, −y)
• n

• n
• zⁿ = (x + y)φ(x, −y)

x + y

n^nm−1

m

• x + y = n^mn−1aⁿ
• φ(x, −y) = nαⁿ

• a
• α
• n