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This paper illustrates that the perfect cuboid problem, which is also known as perfect Euler brick problem, can be easily and conveniently represented by a prism instead of a cuboid. This will make the concept simpler and easier to understand. It eliminates the use of solid diagonal of the cube, which used to be a hidden line. It makes the same lin...
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... Adding to all those, this paper gives much more fundamental equation which is applicable to all types of triangles. It is important to note that perfect cuboid problem [7] is different, as it deals with integers only. Figure 1 shows a right angled triangle in which a = 3 units, b = 4 units and c = 5 units. ...
This paper establishes a basic equation a n + b n = c n applicable for any triangle, having a, b and c as the sides with 'c' being the longest side and 'n' is a number varying from 1 to infinity. Here, a, b, c and n need not always be integers. It also arrives at a relation between largest angle θ (opposite to the longest side 'c') and sides of the triangle with the equation based on cosine rule.The paper graphically and mathematically illustrates the relation between the angle θ and 'n', for different values of 'n' and 'r' (where 'r' is the ratio of sides b/a) for the range of both 'n' and 'r' varying from 1 to infinity. The paper also shows that Pythagoras theorem is a particular case of the above fundamental equation, when n = 2. The paper clearly illustrates with an example that the above fundamental equation is valid even when any one (or two or all) of the sides a, b or c will become non-integer values for all powers of n > 2. This gives a clear way of understanding the Fermat's Last Theorem. MSC: 51P99, 60A99.
