Number Theory (1)

**NonCommAlg** · November 21st 2009, 09:56 PM

This interesting question is taken from an Iranian (

) competetion (high school level): (Have fun!)

Suppose $x,y,z$ are positive integers with $xy=z^2+1.$ Prove that there exist integers $a,b,c,d$ such that $x=a^2+b^2, \ y=c^2+d^2,$ and $z=ac+bd.$

**simplependulum** · November 21st 2009, 11:45 PM

you mean

$(a^2 + b^2 )(c^2 + d^2 ) = (ac + bd )^2 + 1 \implies$

$(ad - bc )^2 = 1$

$ad - bc = 1$ or $bc - ad = 1$

I remember that If $(A,B) = 1$ , then there exists integers

$A,B$ such that $Ax + By = 1$ , $(x,y) \in \mathbb{N}$

Is it correct if $(a,b) = (a,c) = (b,d) = (c,d) = 1$

there also exists integers $a,b,c,d$ satisfying

$ad - bc = 1$ or $bc - ad = 1$ ?

For exmaple , set $a = 5 , c = 3$

$5d - 3b = 1$ , i obtain $b = 3 + 5t , d = 2 + 3t$

let $t = 0$

therefore

$x = a^2 + b^2 = 25 + 9 = 34$

$y = c^2 + d^2 = 9 + 4 = 13$

$z = ac + bd = 15 + 6 = 21$

$34 \times 13 = 442 = z^2 + 1$

It is also true when $t = 1$

$a = 5 , b= 8 , c=3 , d=5$

$(5^2 + 8^2 )( 3^2 + 5^2 ) = \left ( (5)(3) + (8)(5) \right)^2 + 1$