International Journal of Algebra and Statistics Volume 4: 1(2015), 39–41
Published by Modern Science Publishers Available at: http://www.m-sciences.com
A proof of Fermat’s last theorem using elementary algebra James E. Josepha,b a Retired,
Department of Mathematics, Howard University, Washington, DC 20059, USA. b 35 E Street NW #709, Washington, DC 20001, USA. (Received: 8 February 2015; Accepted: 31 March 2015)
Abstract. In 1995, A. Wiles [1, 2] announced a proof of Fermat’s Last Theorem, which is stated as follows:
If π is an odd prime and x, y, z are relatively prime positive integers, then zπ , xπ + yπ . In this note, a simpler proof of this theorem is offered. It is proved that if π is an odd prime and x, y, z are positive integers satisfying zπ = xπ + yπ , then x, y, and z are each divisible by π. The proof offered by Wiles uses cyclic groups. It is the purpose of this note to offer a proof of this result, with techniques within the grasp of Fermat.
The special case z4 = x4 + y4 is impossible for relatively prime integers x, y and z [3]. A new proof of this result is offered in this paper by showing that if z4 = x4 + y4 for positive integers x, y, z, then x, y and z are each even. Initially, it is shown that if x, y, z, are relatively prime positive integers, π is an odd prime, zπ , xπ + yπ . If x and π are positive integers, the notation x ≡ 0 (mod π) will mean x is divisible by π. Let C(π, k) represent the kth coefficient of the binomial expansion of (x+ y)π ; if π is prime, then C(π, k) ≡ 0 (mod π) for every 1 < k < π; C(4, k) ≡ 0 (mod 2) for every 1 < k < 4. Theorem 0.1. If x, y, z are positive integers, π an odd prime and xπ + yπ = zπ , then x ≡ 0 (mod π), y ≡ 0 (mod π), z ≡ 0 (mod π). Theorem 0.1 is arrived at through the first two Lemmas. Lemma 0.2. If x, y, z are positive integers, π a prime, and zπ = xπ + yπ , then: (1) (2) (3) (4) (5) (6) (7) (8)
(x + y)π − zπ ≡ 0 (mod π), (z − x)π − yπ ≡ 0 (mod π), (z − y)π − xπ ≡ 0 (mod π), x + y − z ≡ 0 (mod π), (x + y)π − zπ ≡ 0 (mod π2 ), (z − y)π − xπ ≡ 0 (mod π2 ), (z − x)π − yπ ≡ 0 (mod π2 ), x + y − z , 0.
Proof. Using the equation zπ = xπ + yπ , statements (1), (2), and (3) are obvious; (4), (5), (6), and (7) come from the following three equations: (x + y)π − zπ =
π−1 X
C(π, k)(x + y − z)π−k zk .
0
2010 Mathematics Subject Classification. 11Yxx Keywords. Fermat Email address:
[email protected],
[email protected] (James E. Joseph)
(1)
James E. Joseph / Int. J. of Algebra and Statistics 4 (2015), 39–41
(z − y)π − xπ = (z − x − y + x)π − xπ =
π−1 X
40
C(π, k)(z − x − y)π−k xk .
(2)
C(π, k)(z − x − y)π−k yk .
(3)
0
π
π
π
π−1 X
π
(z − x) − y = (z − x − y + y) − y =
0
(8) comes from x, y being positive since π
π
π
π
π
(x + y) − z = 0 ⇒ x + y = z = (x + y) = x +
π−1 X
C(π, k)x
π−k k
π
y +y ⇒
1
π−1 X
C(π, k)xπ−k yk = 0 ⇒ xy = 0.
1
Lemma 0.3. If π is an odd prime and x, y, z are positive integers such that zπ = xπ + yπ , then: (1) z ≡ 0 (mod π), (2) y ≡ 0 (mod π), (3) x ≡ 0 (mod π). Proof. (1) (x + y − z)π
= (x + y)π +
π−1 X (−1)k (x + y)π−k zk − zπ , 1
(x + y − z)π
= (x + y − z + z)π +
π−1 X
C(π, k)(−1)k (x + y)π−k zk − zπ
1
= (x + y − z + z)π +
π−1 X
C(π, k)(−1)k (x + y − z)π−k zk − zπ ≡ 0 (mod π2 ),
1
0
0
= =
π−1 X
C(π, k)(x + y − z)
=
z
≡
z +
π−1 X
1
1
π−1 X
π−1 X
C(π, k)(x + y − z)π−k zk +
1
0
π−k k
π−1 X 1
C(π, k)(−1)k (x + y)π−k zk ≡ 0 (mod π2 ), C(π, k)(−1)k (x + y − z + z)π−k zk ≡ 0 (mod π2 ),
1
π−k X k π−k−v v C(π, k)(−1) C(π − k, v)(x + y − z) z zk ≡ 0 (mod π2 ), 0
0 (mod π),
(2) π
(z − x − y)
=
π X
C(π, k)(−1)k (z − x)π−k yk ≡ 0 (mod π2 ),
0
0
=
π−1 X 1
y
≡
π−k X π−k−v v C(π, k)(−1) C(π − k, v)(z − x − y) y yk ≡ 0 (mod π2 ),
0 (mod π).
k
1
James E. Joseph / Int. J. of Algebra and Statistics 4 (2015), 39–41
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(3) x + y − z ≡ 0 (mod π). Fermat’s Last Theorem: If π is an odd prime and x, y, z are relatively prime positive integers, then zπ , xπ + yπ , also z4 , x4 + y4 . Proof. If π is an odd prime. then z ≡ 0 (mod π); y ≡ 0 (mod π), x ≡ 0 (mod π), so x ≡ 0 (mod π), y ≡ 0 (mod π), z ≡ 0 (mod π), if zπ = xπ + yπ ; with the same arguments as above, and the fact that (x + y)4 − z4 ≡ 0 (mod 2), (z − y)4 − x4 ≡ 0 (mod 2), (z − x)4 − y4 ≡ 0 (mod 2), leads to z ≡ 0 (mod 2), y ≡ 0 (mod 2), x ≡ 0 (mod 2), it follows that if z4 = x4 + y4 , x ≡ 0 (mod 2), z ≡ 0 (mod 2), y ≡ 0 (mod 2). References [1] A. Wiles, Modular ellipic eurves and Fermat’s Last Theorem, Ann. Math. 141 (1995), 443-551. [2] A. Wiles and R. Taylor, Ring-theoretic properties of certain Heche algebras, Ann. Math. 141 (1995), 553-573. [3] H. Edwards, Fermat’s Last Theorem:A Genetic Introduction to Algebraic Number Theory, Springer-Verlag, New York, (1977).
