Last year (2016), in the Interesting Engineering article entitled “Revolution in the Pythagoras’ Theorem?”, Dr. Luis Teia presented the proof of the Pythagoras’ theorem in 3D. This year, Teia explains in his recent (Feb 2017) peer reviewed paper, entitled *Fermat’s Theorem – a Geometrical View* published in the Journal of Mathematics Research, how this 3D understanding of the Pythagoras’ Theorem provided the geometrical foundation to prove Fermat’s Last Theorem. Fermat’s Last Theorem, also known as Fermat’s conjecture, is more than just about triples, it is about the fundamental nature of an integer number, and it’s mathematical and geometrical meaning. It raises the philosophical question: *What is a unit?* In the language of mathematics, a unit is defined by the number 1. In the language of geometry, a unit is defined by an element of side length one. A perspective of a problem depends on the language we use to observe it, and a change in perspective is often all it takes to see the solution.

**What is Fermat’s Theorem? **

Fermat’s Last Theorem questions not only what is a triple, but more importantly, what is an integer in the context of equations of the type X^{n} +Y^{n} = Z^{n}. The picture below shows in a pictorial way the difference between the Pythagoras’ theorem and Fermat’s Last theorem. These two are sometimes confused. Fermat’s Last Theorem is a mathematical conjecture about integer numbers, while the 3D Pythagoras’ theorem is a mathematical and geometrical proof about real numbers. The Pythagoras’ theorem in 1D is the principle of summation (i.e., X+Y=Z). In it, all integers form triples [e.g., 1+2=3 forms the 1D triple (1,2,3) while 3+4=7 forms (3,4,7)]. In the middle is the well-known Pythagoras’ theorem in 2D, where only some integers form triples [e.g., 3^{2}+4^{2}=5^{2} forms the 2D triples (3,4,5)]. Fermat’s Last Theorem states that no triples can be found for Pythagoras’ theorem in 3D, or for any higher dimension.

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The Pythagoras’ Theorem in 1D, 2D and 3D, and Fermat’s Last Theorem [Image Source: *Teia*]

**The 3D Pythagoras’ Theorem**

The Pythagoras’ theorem in 1D is governed by lines, while in 2D by squares (see picture below). Just like squares appear naturally when transforming the Pythagoras’ theorem from 1D to 2D, octahedrons also appear naturally when transforming the Pythagoras’ theorem from 2D to 3D. As shown by Dr. Teia (in his book published in 2015), the 3D Pythagoras’ theorem is governed by octahedrons. Therefore, any number (real or integer) within the Pythagoras’ theorem is expressible geometrically by a line in 1D, a square in 2D and an octahedron in 3D. *How does this geometrical notion affect our understanding of integers, and more importantly of triples?*

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The 1D, 2D and 3D Pythagoras’ Theorem [Image Source: *Teia*]

**Hypothesis**

** **The hypothesis of this new proof is that a triple only exists, if all integer elements within that triple also exist [e.g., 1, 2, 3 for the 1D triple (1,2,3), and 3, 4, 5 for the 2D triple (3,4,5)]. In turn, an integer element only exits if it obeys two conditions: it satisfies the Pythagoras’ theorem of the respective dimension (Condition 1), and it can be completely successfully split into multiple unit scalars (Condition 2). One can therefore hypothesize that integer elements do not exist if either Condition 1 or 2 is not satisfied. By consequence, if the integer does not exist, then the associated triples also do not exist.

**The Geometrical Integer**

** **Integers are clear multiples of a unit. The unit line, or line of length 1, is the fundamental geometric scalar that composes all integer elements in the 1D Pythagoras’ universe. Likewise, the unit square, or square of side 1, is the fundamental geometric scalar that composes all integer elements in the 2D Pythagoras’ universe. Generally, one can conclude that in order for an integer element to exist, it needs to be completely split into multiples of the fundamental unit scalar particular to that dimension (i.e., unit line in 1D or unit square in 2D). In 3D, despite octahedrons validating the 3D Pythagoras’ Theorem (satisfying Condition 1), an octahedron with side integer N is not a multiple of unit octahedrons, as tetrahedrons appear in the middle (refer to figure below right) [not satisfying Condition 2]. Therefore, geometrical integers do not exist in the 3D domain of the Pythagoras’ theorem, and neither do their triples. This satisfies Fermat’s theorem for three dimensions.

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The geometrical definition of integers in 1D, 2D and not in 3D [Image Source: *Teia*]

**Higher Dimensions**

The geometrical interdependency between integers in 1D and 2D suggests that all integers of higher dimensions are built, and hence are dependent, on the integers of lower dimensions (e.g. squares are built with lines). This interdependency coupled with the absence of integers in 3D suggests that there is no integer above n > 2, and therefore there is also no triples that satisfy X^{n} + Y^{n} = Z^{n} for n > 2.

**Conclusion**

The geometrical solution to Fermat’s riddle doesn’t come from the notion of triples, but rather from the notion of integers. If integers don’t exist, then neither can triples. Alas, the centennial elusiveness of the proof results from the repetitive usage of available “tools”, rather than inventing new tools (the 3D Pythagoras’ theorem) to find the solution. The simplicity of this geometrical proof (founded on the absence of integers within the domain of the Pythagoras’ theorem for dimensions above 2D) makes us wonder whether this is not the famous “elegant solution” that Fermat spoke of, of which he left no other records except a written note saying:

“I have discovered a truly remarkable proof of this theorem, which this margin is too small to contain.”

–Pierre de Fermat (1665)

As for Dr. Luis Teia, his next challenge will be to explain the geometrical meaning of the formula on partitions from the mathematician Srinivasa Ramanujan.

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