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=m^{2}n^{2},...(2) c= m^{2}+n^{2} ;....(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
