Question Number 189970 by Rupesh123 last updated on 25/Mar/23
Answered by Rasheed.Sindhi last updated on 25/Mar/23
$${x},{y},{x}+{y},{x}−{y}\:\in\mathbb{P}\:\:;\:{x},{y}=? \\ $$$$\bullet{If}\:\:{x}\:{and}\:{y}\:{were}\:{both}\:{odd}\:{primes} \\ $$$$\Rightarrow\mathrm{2}\:\mid\:\left({x}+{y}\right)\Rightarrow{x}+{y}\:\notin\mathbb{P}\:{or}\:{x}+{y}=\mathrm{2} \\ $$$${x}+{y}=\mathrm{2}\:{has}\:{no}\:{solution}\:{in}\:{prime} \\ $$$$\bullet{If}\:{x}={y}=\mathrm{2}\Rightarrow\:{x}+{y},{x}−{y}\:\notin\mathbb{P} \\ $$$${Hence}\:{one}\:{of}\:{x}\:{and}\:{y}\:{is}\:{even}\:{prime} \\ $$$${and}\:{the}\:{other}\:{is}\:{odd}\:{prime}. \\ $$$$\bullet{Assuming}\:{x}−{y}>\mathrm{0}\:\:\Rightarrow{y}=\mathrm{2} \\ $$$$\bullet\:{x}+\mathrm{2},{x},{x}−\mathrm{2}\:{are}\:{twin}\:{primes}\:\left({having}\right. \\ $$$$\left.{difference}\:\mathrm{2}\right) \\ $$$$\bullet{Only}\:{three}\:{consecutive}\:{twin}\:{primes}\:{are} \\ $$$$\mathrm{3},\mathrm{5},\mathrm{7} \\ $$$$\:\:\:{x}+\mathrm{2}=\mathrm{7},{x}=\mathrm{5},{x}−\mathrm{2}=\mathrm{3} \\ $$$$\bullet{x}=\mathrm{5},{y}=\mathrm{2} \\ $$$$ \\ $$
Commented by Rupesh123 last updated on 25/Mar/23
Excellent!