Primitive Pythagorean triples with the least, same hypotenuse or side

May 2015
488
29
Arlington, VA
What are the primitive Pythagorean triples with the least, same hypotenuse or side?
 
May 2015
488
29
Arlington, VA
Say, where hypotenuses of two differing primitive Pythagorean triples are equal.
 
Feb 2010
720
153
$\begin{align} 1105^2 &=47^2+1104^2\\ &=264^2+1073^2\\ &=576^2+943^2\\ &=744^2+817^2 \end{align}$

It took google 0.62 seconds to find these four.
 
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May 2015
488
29
Arlington, VA
mrtwhs,

What is the smallest such hypotenuse with just two pairs of legs?
 
Last edited:
Feb 2010
720
153
Don't know if it is the smallest but google gives $33^2+56^2 = 16^2 +63^2 = 65^2$
 

skipjack

Forum Staff
Dec 2006
21,315
2,383
It is the smallest. The smallest pair with a leg in common is ((5, 12, 13), (12, 35, 37)).
 
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