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\((a,b,c,d \in \mathbb{N}^+ \; ab=cd) \Rightarrow (a^2+b^2+c^2+d^2 \text{ is not prime})\)
05-01-2010, 10:36 PM
Post: #1
\((a,b,c,d \in \mathbb{N}^+ \; ab=cd) \Rightarrow (a^2+b^2+c^2+d^2 \text{ is not prime})\)
Let \(\gcd(a,c)=x, \ \gcd(b,d)=y.\) then \(a=xa_0, \ c=xc_0, \ b=yb_0, \ d=yd_0,\) where \(\gcd(a_0,c_0)=\gcd(b_0,d_0)=1.\) thus, since \(ab=cd\), we must have \(a_0=d_0\) and \(b_0=c_0\).

therefore \(a^2+b^2+c^2+d^2=(x^2+y^2)(a_0^2+b_0^2). \ \Box\)

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