Primality of Fermat Numbers

Remark 5.1. Notice that the number % MathType!MTEF!2!1!+-% feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0JeabeqaamaabaabaaGcbaGaamOramaaBa % aaleaacaWGTbaabeaakiabg2da9abeqaamaabaabaaGcbaGaamOramaaBa % aaleaacaWGTbaabeaakiabg2da9iaaikdadaahaaWcbeqaaiaaikda % daahaaadbeqaaiaad2gaaaaaaOGaey4kaSIaaGymaiaaywW7caWGMb % Gaam4BaiaadkhacaaMc8UaamyBaiabg2da9iaaicdacaGGSaGaaGym % aiaacYcacaaIYaGaaiilaiablAcilbaa!4A9D!\[{2^{{2^3}}+1} +{2^8}+1 $$ is prime, but the numbers 23 + 1 and % MathType!MTEF!2!1!+-% feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0J0-yqaqpeqaamaabaabaaGcbaGaamOramaaBa % aaleaacaWGTbaabeaakiabg2da9iaaikdadaahaaWcbeqaaiaaikda % daahaaadbeqaaiaad2gaaaaaaOGaey4kaSIaaGymaiaaywW7caWGMb % Gaam4BaiaadkhacaaMc8UaamyBaiabg2da9iaaicdacaGGSaGaaGym % aiaacYcacaaIYaGaaiilaiablAcilbaa!4A9D!\[2^{{2^8}}+1 $$ are composite (cf. Appendix A). This example shows that if 2n prime, then % MathType!MTEF!2!1!+-% feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaamOramaaBa % aaleaacaWGTbaabeaakiabg2da9iaaikdadaahaaWcbeqaaiaaikda % daahaaadbeqaaiaad2gaaaaaaOGaey4kaSIaaGymaiaaywW7caWGMb % Gaam4BaiaadkhacaaMc8Uaa9iaaicdacaGGSaGaaGym % aiaacYcacaaIYaGaaiilaiablAcilbaa!4A9D!\[2^{{2^n}}+1 $$ need not be prime and vice versa (see [Sierpiński, 1970, Problem 141]).