Pythagorean Puzzles

 (b')  (m') (n' ) -- (n" ) -- (a')  (c')  (a")  (c")  (b") Area  under (b',a',c'): Area  under (b",a",c"): -(*m")  = (*put even integers till u get blue color value as an integer . There may be multiple even  integers to give multiple blue value) Can the long side of one Pythagorean triplet be the short side of another Pythagorean triplet? Ans: If the short sides of the above triplet are a,b and long side c, they must satisfy a=2mn...(1), b=m2-n2,...(2) c= m2+n2 ;....(3) where m,n are integers. The long side is always an odd no. So to become short side, it should satisfy (2) only because if it satisfies..(1), it has to be an even no. which it is not. b'=c=m'2-n'2 .or n'2  =b'+ m'2  . b' is known. Take integer values of m' to find out  for which value of m', n' is an integer. Similarly, n'2  =b'- m'2  . b' is known. Take integer values of m' to find out  for which value of m', n' is an integer. Example1: Let us start with triplet (3,4,5)(b,a,c) with b=3,a=4. press submit -> c=5 which is an integer. now b1=c=5. Put b'=5, start with m'=1,2,3..... till n' becomes an integer. For m'=2, it becomes integer with value n'=3 (n'2  =b'+ m'2) value of a'=12 (2n'm'), c'=13(m'2+n'2). Similarly, one can try with a' & c' arrived at by n'2  =b'- m'2 Now put the value of a' in a1=12. you get the triplet (5,12,13) where b1=5,a1=12,c1=13. Now again put b'=c1=13, start with m'=1,2,3... till n' becomes an integer. For m'=6, it becomes integer. value of a' =84,c'=85. Now put again the current value of a' in a2=84. You get the triplet (13,84,85) where b2=13,a2=84 & c2=85. Now again put b'=c2=85, start with m'=1,2,3... till n' becomes an integer. For m'=6, it becomes integer. There is another triplet (36,77,85). value of a' =132,c'=157. Now put again the current value of a' in a3=132. You get the triplet (85,132,157) where b3=85,a3=132 & c3=157.Now again put b'=c2=157, start with m'=1,2,3... till n' becomes an integer. We tried up to m'=50, but it did not become an integer. Example 2 : Suppose there is Pythagorean triplet whose one of the short sides is a long side of another Pythagorean triplet. Find the triplet. Enter the long side in b'. put integers in m' till u get a value of n" acceptable. click submit. Then put even integer values in m" till the blue colored figure is an integer. Corresponding value of (b",a",c") are the triplets Try with  b' =13, m'=2. Then put m" =2. You get (4,3,5) -- (  b    )- (   a   )-(  c     ) -- (  b1    )- (   a1   )-(  c1     ) -- (  b2    )- (   a2   )-(  c2     ) -- (  b3    )- (   a3   )-(  c3     ) -- (  b4    )- (   a4   )-(  c4     ) Put P While exploring Pythagorean triplets, one may find 2 or multiple triplets having one number identical i.e. either the long side or any one of the short sides. Exa- (13,84,85) & (36,77,85) which have the long side 85 same in both triplets. In such cases, we can call the triplets degenerate with respect to that specific side.