# Generating special numbers [closed]

Here are three numbers that are related to one another: 9841, 8591, 4800

How are these numbers related and how are they generated?

They form a

Pythagorean triple, i.e. $$9841^2 = 8591^2 + 4800^2$$. When used as the lengths of the sides of a triangle, the triangle has a right angle.

The general formula is:

The triples $$(a,b,c)$$ such that $$a^2+b^2=c^2$$ are of the form:
$$c=k(u^2+v^2) \\ b=2kuv \\ a=k(u^2-v^2)$$ Where $$u,v$$ are coprime and not both odd. Cases where $$k=1$$ are called primitive triples, and they are such that $$a,b,c$$ have no common factor.

This case is a primitive triple with $$u=96$$ and $$v=25$$.

• Your answer is partially correct. How did you obtain the values u and v? – Vassilis Parassidis Jan 16 at 4:39
• @VassilisParassidis After checking that there is no common factor (so $k=1$) I just used $c+a=2u^2$ and $c-a=2v^2$. – Jaap Scherphuis Jan 16 at 4:45
• Actually, even when $u$ and $v$ are both odd or non-coprime, they will still be valid Pythagorean triples (although they may be non-primitive - having $\gcd(a,b,c)>1$ - even when $k=1$). – trolley813 Jan 16 at 9:23
• @trolley813 That's true, but unfortunately you cannot get all non-primitive triangles that way, so you have to put in the factor $k$, and then it is neater to ensure that is the only common factor between them. – Jaap Scherphuis Jan 16 at 10:14
• @VassilisParassidis This seems very terribly “what am I thinking of?” more than an actual puzzle. There’s no clues toward the unique formulation you’ve described that would help a solver arrive at that particular pattern. Puzzles should be forward solvable from their clues, not merely be verifiable once the solution is known. How, then, is this a puzzle? – Rubio Jan 19 at 7:38