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A solution to this problem is an example of a Kirkman triple system, [3] which is a Steiner triple system having a parallelism, that is, a partition of the blocks of the triple system into parallel classes which are themselves partitions of the points into disjoint blocks.
A primitive Pythagorean triple is one in which a, b and c are coprime (that is, they have no common divisor larger than 1). [1] For example, (3, 4, 5) is a primitive Pythagorean triple whereas (6, 8, 10) is not. Every Pythagorean triple can be scaled to a unique primitive Pythagorean triple by dividing (a, b, c) by their greatest common divisor ...
The problem asks if it is possible to color each of the positive integers either red or blue, so that no Pythagorean triple of integers a, b, c, satisfying + = are all the same color. For example, in the Pythagorean triple 3, 4, and 5 ( 3 2 + 4 2 = 5 2 {\displaystyle 3^{2}+4^{2}=5^{2}} ), if 3 and 4 are colored red, then 5 must be colored blue.
To find the primitive Pythagorean triple associated with any such value t, compute (1 − t 2, 2t, 1 + t 2) and multiply all three values by the least common multiple of their denominators. (Alternatively, write t = n / m as a fraction in lowest terms and use the formulas from the previous section.)
Plimpton 322 is a Babylonian clay tablet, believed to have been written around 1800 BC, that contains a mathematical table written in cuneiform script.Each row of the table relates to a Pythagorean triple, that is, a triple of integers (,,) that satisfies the Pythagorean theorem, + =, the rule that equates the sum of the squares of the legs of a right triangle to the square of the hypotenuse.
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Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
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