The Pythagorean theorem is a widely used relationship in mathematics and physics, as it relates the lengths a and b of two sides of a right triangle to its hypotenuse c via the formula a + b = c2. The mathematical proof assumes the triangle lies on a at Euclidean space [1]. Because general relativity causes matter to curve space [2], one expects the Pythagorean theorem to not precisely hold in ...

The concept of information distance in non-commutative setting is re-considered. Additive information, such as Kullback-Leibler divergence, is defined using convex functional with gradient having the property of homomorphism between multiplicative and additive subgroups. We review several geometric properties, such as the logarithmic law of cosines, Pythagorean theorem and a lower bound given b...

PYTHAGOREAN THEOREM AND CORRESPONDING FUNCTIONAL EQUATIONS

We investigate some combinatorial aspects of the “Pythagorean triple system”. Our motivation is the following question: Is it possible to color the naturals with finitely many colors so that no Pythagorean triple is monochromatic? This question is open even for two colors. A natural approach is to search for a nonbipartite triple system that can be realized as a family of Pythagorean triples. S...

Is it possible to color the naturals with finitely many colors so that no Pythagorean triple is monochromatic? This question is even open for two colors. A natural strategy is to show that some small nonbipartite triple systems cannot be realized as a family of Pythagorean triples. It suffices to consider partial triple systems (PTS’s), and it is therefore natural to consider the Fano plane, th...

We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a non-Euclidean geometry, finding three integers such that the difference of the squares of any two is a square, and the problem of finding rational points on an algebraic surface in algebraic geometry. We will also reinterpret Euler’s work on the second problem with a mo...