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If several such values happen to lie close together, the corresponding parabolas approximately coincide, and the triples cluster in a narrow parabolic strip. For instance, 38 2 = 1444, 2 × 27 2 = 1458, 3 × 22 2 = 1452, 5 × 17 2 = 1445 and 10 × 12 2 = 1440; the corresponding parabolic strip around n ≈ 1450 is clearly visible in the scatter ...
An arbitrary coefficient can be placed in front of the x-value on either m or n, which causes the resulting equation to systematically "skip" through the triples. For example, consider the triple [20,21,29], which can be calculated from the Euclid equations with values m = 5 and n = 2.
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.)
Pythagoras is credited with having devised the tetractys, [80] an important sacred symbol in later Pythagoreanism. [ 81 ] [ 82 ] 1st century treatises of Philo and Nicomachus popularised the mystical and cosmological symbolism Pythagoreans attributed to numbers .
A Pythagorean quadruple is called primitive if the greatest common divisor of its entries is 1. Every Pythagorean quadruple is an integer multiple of a primitive quadruple. The set of primitive Pythagorean quadruples for which a is odd can be generated by the formulas = +, = (+), = (), = + + +, where m, n, p, q are non-negative integers with greatest common divisor 1 such that m + n + p + q is o
Pythagoras constant [4] 1.41421 35623 73095 04880 ... Decimal representations are rounded or padded to 10 places if the values are known. Name Symbol Set
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The Boolean Pythagorean triples problem is a problem from Ramsey theory about whether the positive integers can be colored red and blue so that no Pythagorean triples consist of all red or all blue members.