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The Angular Approach to Fermat’s Last Theorem,
(By G. Montgomery, Email: [email protected])
Abstract
“The Angular Approach to Fermat’s Last Theorem” argues that since all possible solutions to xn + yn = zn, require
that both x & y are less than z and that (x + y) is greater than z, a triangle can be used to represent every possible
solution. Given that all triangles can be classified as belonging to only one of three groups where angle Z is acute,
obtuse or right angled, the problem is simplified to considering how the classification of angle Z changes as we
increment n until we can no longer form a new triangle. This paper will show that regardless of which group angle
Z initially belongs to, a right angled Pythagorean Triple can never be created for n > 2 thereby confirming
Fermat’s Last Theorem: xn + yn ≠ zn for n > 2 (x, y, z & n positive integers).
x values
Z< 90º
Z = 90º
y values
Z > 90º
z values
Fig 1: All potential solutions for x, y & z form an angle Z which is: Acute, Obtuse or Right Angled
Whilst there are an infinite number of potential triangles, angle Z can be classified as: acute, obtuse or right angled
and by using the Law of Cosines (z2 = x2 + y2 – 2 x y Cos Z), we can extrapolate what happens when n ≥ 2. As we
increment n we consider the new values formed and if a new triangle cannot be formed or an obtuse angle is
formed we will see that no equality can ever exist for these particular values of x, y & z. If an acute angle is formed
then we increment n again until either an obtuse angle or a right angle is formed. It will be shown however, that a
right angled Pythagorean Triple can never be formed for n > 2 and it is this result that confirms Fermat’s Last
Theorem.
Group 1: When the initial angle Z is obtuse (Z > 90º), Cos Z is negative and z2 > x2 + y2. As we increase n, z2 will
always be greater than (x2 + y2) and equality can never exist for n ≥ 2.
Group 2: When the initial angle is right angled, Z = 90º and Cos 90º = 0, x2 + y2 = z2, giving the Pythagorean
equality for right angled triangles. Clearly, these equalities are only valid for n =2.
Group 3: When the initial angle is acute Z < 90º, Cos Z is positive and z2 < x2 + y2. In this case, however, as we
increase n, zn will grow more quickly than xn + yn and angle Z will become less acute, eventually becoming obtuse
as we continue to increase n. There is the possibility that during the transition from acute to obtuse a Pythagorean
right angled equality will be produced at some value of n > 2. Given that x, y, z & n are positive integers any right
angled triangle produced for n > 2 would have integer sides, constitute a Pythagorean Triple and would have to
obey all the laws for right angled triangles. The new triangle would coincide with one of the known triangles such
that xn + yn = zn would map into a2 + b2 = c2. The hypotenuse of every Primitive Pythagorean Triple is a prime
number and for every Non Primitive Triple the hypotenuse is an integer multiple of a prime number. If we equate
zn = (mc)2 where c is a prime number and m is a integer, we obtain z = m2/n c2/n. However, as c is a prime number
c2/n cannot have an integer root for n > 2 and therefore z can never be an integer. Pythagorean Triples have many
properties that have to be satisfied, some of which are listed below (source: Wikipedia).
Clearly, substitution of a, b & c, with xn/2, yn/2 & zn/2 shows that the properties of a Pythagorean Triple can only be
satisfied for n = 2 and that right angled (integer) triangles can not exist for n > 2. This result confirms Fermat’s
Last Theorem: xn + yn ≠ zn for n > 2.
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Properties of a Pythagorean Triple:
* a + b = c + 2*sqrt{(c - a)(c - b)/2}
* (c − a)(c − b)/2 is always a perfect square.
* Exactly one of a, b is odd; c is odd.
* The area (A = ab/2) is an integer.
* Exactly one of a, b is divisible by 3.
* Exactly one of a, b is divisible by 4.
* Exactly one of a, b, c is divisible by 5.
* Exactly one of a, b, (a + b), (b − a) is divisible by 7.
* All prime factors of c are primes of the form 4n + 1.
* At most one of a, b is a square.
* Every integer greater than 2 that is not congruent to 2 mod 4 is part of a primitive Pythagorean triple. In other
words integers greater than 2 which are not of the form 4n + 2.
* Every integer greater than 2 is part of a primitive or non-primitive Pythagorean triple, for example, the integers
6, 10, 14, and 18 are not part of primitive triples, but are part of the non-primitive triples 6, 8, 10; 14, 48, 50 and
18, 80, 82.
* There exist infinitely many primitive Pythagorean triples whose hypotenuses are squares of natural numbers.
* There exist infinitely many primitive Pythagorean triples in which one of the legs is the square of a natural
number.
* There exist infinitely many Pythagorean triples in which the hypotenuse and the longer of the two legs differ by
exactly one (such triples are necessarily primitive). Generalization: For every odd integer j, there exist infinitely
many primitive Pythagorean triples in which the hypotenuse and the even leg differ by j 2.
* There exist infinitely many primitive Pythagorean triples in which the hypotenuse and the longer of the two legs
differ by exactly two. Generalization: For every integer k > 0, there exist infinitely many primitive Pythagorean
triples in which the hypotenuse and the odd leg differ by 2k2.
* If j and k are odd positive integers, not necessarily unequal, there is exactly one primitive Pythagorean triple with
a + j2 = c = b + 2k.
* The hypotenuse of every primitive Pythagorean triangle exceeds the even leg by the square of an odd integer j,
and exceeds the odd leg by twice the square of an integer k > 0, from which it follows that:
* There are no primitive Pythagorean triples in which the hypotenuse and a leg differ by a prime number greater
than 2.
* The sum of the hypotenuse and the even leg of a primitive Pythagorean triple is the square of an odd number,
and the arithmetic mean of the hypotenuse and the odd leg is a perfect square.
* For each natural number n, there exist n Pythagorean triples with different hypotenuses and the same area.
* For each natural number n, there exist at least n different Pythagorean triples with the same leg a, where a is
some natural number
* For each natural number n, there exist at least n different triangles with the same hypotenuse.
* In every Pythagorean triple, the radius of the incircle and the radii of the three excircles are natural numbers.
(Actually the radius of the incircle can be shown to be r = n(m − n) )
* There are no Pythagorean triplets in which the hypotenuse and one leg are the legs of another Pythagorean triple.
Bibliography:
Fermat’s Last Theorem by Simon Singh; published by Harper Perennial 2005.
Thee Last Problem by E.T. Bell; published by London Victor Gollancz Ltd 1962.
The properties of a Primitive Pythagorean Triple, Wikipedia