Posted on March 26th, 2012, by essay

A set of three integer numbers {a, b, c} is called a **Pythagorean triple**, if the sum of squares of two of these numbers equals to the square of the third number. These triples are one of first applications of the number theory, since they are directly related to Pythagoras theorem. The theorem states the following: for a right triangle, the square of hypotenuse is equal to the sum of squares of the legs of this triangle. The numbers satisfying the relation Β and having no common divisor beside 1 are called primitive Pythagorean triples. They are called primitive, because other triples might be obtained from these triples by multiplying them all to a certain coefficient.

The most famous example of a Pythagorean triple is the set of numbers {3, 4, 5}. It is not easy to find other triples empirically. However, there is a formula for obtaining Pythagorean triples: , where s and t are randomly selected integer numbers and s is the greater of them. Let us generate five Pythagorean triples by selecting different s and t values.

Triple 1. s = 2, t = 1; a = 2*2*1=4; b = s^{2} – t^{2}=4-1=3; c = s^{2} + t^{2}=4 + 1 = 5; Pythagorean triple: {4, 3, 5}; check: 4^{2} + 3^{2}=16+9=25=5^{2}

Triple 2. s = 3, t = 2; a = 2*3*2=12; b = s^{2} – t^{2 }= 9 4 = 5; c = s^{2} + t^{2}=9 + 4 = 13; Pythagorean triple: {12, 5, 13}; check: 12^{2} + 5^{2}=144+25=169=13^{2}

Triple 3. s = 4, t = 2; a = 2*4*2=16; b = s^{2} – t^{2}=16-4=12; c = s^{2} + t^{2}=16 + 4 = 20; Pythagorean triple: {16, 12, 20}; check: 16^{2} + 12^{2}=256+144=400=20^{2}

Triple 4. s = 4, t = 3; a = 2*4*3=24; b = s^{2} – t^{2 }= 16 9 = 7; c = s^{2} + t^{2}=16 + 9 = 25; Pythagorean triple: {24, 7, 25}; check: 24^{2} + 7^{2}=576+49=625=25^{2}

Triple 5. s = 5, t = 2; a = 5*2*2=20; b = s^{2} – t^{2 }= 25 – 4=21; c = s^{2} + t^{2}=25 + 4 = 29; set: {20, 21, 29}; check: 20^{2} + 21^{2}=400+441=841=29^{2}

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