MathLabs

Problem 6

Let a>b>c>d>0 a>b>c>d>0 be integers satisfying ac+bd=(b+d+a−c)(b+d−a+c) ac+bd=(b+d+a-c)(b+d-a+c). Prove that ab+cd ab+cd is not prime.
Step 5 of 6: Contradict primality
Previous stepNext step
p prime⟹p(ad+bc)ac+bd∉Z,p\text{ prime}\Longrightarrow\frac{p(ad+bc)}{ac+bd}\notin\mathbb Z,
Detailed analysis

The denominator is between zero and p and cannot divide the smaller factor, but the identity says the fraction is the integer WY squared.