What is the hypothesis and what is the conclusion in of the following implicatio

Question

What is the hypothesis and what is the conclusion in of the following implication .. - The square of the length of the hypotenuse of a right - angled triangle is the sum of the squares of the length of the other side. - all primes are even

Explanation / Answer

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple (PPT) is one in which a, b and c are pairwise coprime. A right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle. The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a2 + b2 = c2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a = b = 1 and c = ?2 is right, but (1, 1, ?2) is not a Pythagorean triple because ?2 is not an integer. Moreover, 1 and ?2 do not have an integer common multiple because ?2 is irrational. Contents [hide] 1 Examples 2 Generating a triple 2.1 Proof of Euclid's formula 2.2 Interpretation of parameters in Euclid's formula 3 Elementary properties of primitive Pythagorean triples 4 Some relationships 5 A special case: the Platonic sequence 6 Geometry of Euclid's formula 7 Spinors and the modular group 8 Parent/child relationships 9 Relation to Gaussian integers 9.1 As perfect square Gaussian integers 10 Relation to twin primes 11 Distribution of triples 12 Generalizations 12.1 Pythagorean quadruple 12.2 Pythagorean n-tuple 12.3 Fermat's Last Theorem 12.4 n ? 1 or n nth powers summing to an nth power 12.5 Heronian triangle triples 13 See also 14 Notes 15 References 16 External links [edit]Examples A scatter plot of the legs (a,b) of the Pythagorean triples with c less than 6000. Negative values are included to illustrate the parabolic patterns in the plot more clearly. There are 16 primitive Pythagorean triples with c ? 100: ( 3 , 4 , 5 ) ( 5, 12, 13) ( 7, 24, 25) ( 8, 15, 17) ( 9, 40, 41) (11, 60, 61) (12, 35, 37) (13, 84, 85) (16, 63, 65) (20, 21, 29) (28, 45, 53) (33, 56, 65) (36, 77, 85) (39, 80, 89) (48, 55, 73) (65, 72, 97) Each one of these low-c points forms one of the more easily recognizable radiating lines in the scatter plot. Additionally these are all the primitive Pythagorean triples with 100 n, m ? n odd, and with m and n coprime. That these formulas generate Pythagorean triples can be verified by expanding a2 + b2 using elementary algebra and verifying that the result coincides with c2. Since every Pythagorean triple can be divided through by some integer k to obtain a primitive triple, every triple can be generated uniquely by using the formula with m and n to generate its primitive counterpart and then multiplying through by k as in the last equation. Many formulas for generating triples have been developed since the time of Euclid. [edit]Proof of Euclid's formula That satisfaction of Euclid's formula by a, b, c is sufficient for the triangle to be Pythagorean is apparent from the fact that for integers m and n, the a, b, and c given by the formula are all integers, and from the fact that A simple proof of the necessity that a, b, c be expressed by Euclid's formula for any primitive Pythagorean triple is as follows.[3] All such triples can be written as (a, b, c) where a2 + b2 = c2 and a, b, c are pairwise coprime, and where b and c have opposite parities (one is even and one is odd). (If c had the same parity as both legs, then if all were even the parameters would not be coprime, and if all were odd then a2 + b2 = c2 would equate an even to an odd.) From we obtain and hence . Then . Since is rational, we set it equal to in lowest terms. We also observe that equals the reciprocal of and hence equals the reciprocal of , and thus equals . Then solving for and gives Since and are fully reduced by assumption, the numerators can be equated and the denominators can be equated if and only if the right side of each equation is fully reduced; given the previous specification that is fully reduced, implying that m and n are coprime, the right sides are fully reduced if and only if m and n have opposite parity (one is even and one is odd) so that the numerators are not divisible by 2. (And m and n must have opposite parity: if both were odd then dividing through by 2 would give the ratio of two odd numbers; equating this ratio to , which is a ratio of two numbers with opposite parities, would give conflicting parities when the equation is cross-multiplied.) So equating numerators and equating denominators, we have Euclid's formula with m and n coprime and of opposite parities. A longer but more commonplace proof is given in Maor (2007)[4] and Sierpinski (2003).[5] [edit]Interpretation of parameters in Euclid's formula Suppose the sides of a Pythagorean triangle are , and , and suppose the angle between the leg and the hypotenuse is denoted as . Then a right triangle with legs and has the angle between the leg and the (not necessarily rational) hypotenuse.[6]

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