Instituttion Name
Pythagorean triple is a set of positive integers ( a, b, c) that fits the rule: a2+b2=c2. The most well-known set is 3,4,5. If we substitute the letters with the digits we can check that the formula is right and this set of numbers is correct. 32+42=52 – 9+16=25.
Since this set of numbers has been discovered different formulas have been invented in order to generate these triples, but the most common formula belongs to Euclid. The formula is a=m2-n2, b-2mn, c=m2+n2. There are 3 restrictions: m-n must be odd and both should be coprime, m>n.
Building triples:
M=3, N=2 A=32-22= 5 B=2*3*2=12 C = 32+22=13
M=4, N=3 A=42-32= 7 B=2*4*3=24 C = 42+32=25
M=5, N=4 A=52-42= 9 B=2*5*4=40 C = 52+42=41
M=6, N=5 A=62-52= 11 B=2*6*5=60 C = 62+52=61
M=7, N=6 A=72-62= 13 B=2*7*6=84 C = 72+62=85
Verification of triples:
52+122=132 25+144=169
72+242=252 49+576=625
92+402=412 81+1600=1681
112+602=612 121+3600=3721
132+842=852 169+7056=7225
I have chosen this method of generating of Pythagorean triples as it is one of the oldest, but at the same time it is the simplest one. I t allows one with the help of elementary algebra both to generate sets and to verify them. Formula requires only two digits, which are supposed to be coprime and the difference between them should be odd. Using this formula one can generate infinite number of Pythagorean triples.
The well-known use of the triples is in Pythagorean Theorem which can be applied in many everyday situations, but mostly in building and architecture.
