Problem
ALG-B1-M02-P014 Primality in Sophie Germain's Expression
#14
★★★☆☆ Level 3 of 5
Find all positive integer pairs \(x,y\) for which \(x^4+4y^4\) is prime.
Use the factorisation of \(x^4+4y^4\) and determine when the first factor can equal \(1\).
We have \(x^4+4y^4=(x^2-2xy+2y^2)(x^2+2xy+2y^2)\).
The first factor is \((x-y)^2+y^2\). For positive integers \(x,y\), it equals \(1\) only when \(y=1\) and \(x=y\), so \(x=y=1\).
Then the number is \(5\), which is prime. In all other cases both factors are greater than \(1\), so the number is composite. Answer: \((1,1)\).
The difficulty increases because the factors must be analysed.