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Question Number 162674 by Mathematification last updated on 31/Dec/21

Answered by mindispower last updated on 31/Dec/21

$$d\mid 2(3n+1)-3(2n+1)\Rightarrow d\mid -1$$$$d=1$$$$orusebizouidentities$$$$-2(3n+1)+3(2n+1)=1$$$$\Rightarrow (2n+1),(3n+1)areprime$$

Answered by mr W last updated on 31/Dec/21

$$gcd(2n+1,3n+1)=1$$$$$$$$prove:$$$$ifgcd(2n+1,3n+1)=k\ne 1$$$$thenthereexistintegersp,qsuchthat$$$$2n+1=kp$$$$3n+1=kq$$$$\Rightarrow n=\frac{kp-1}{2}=\frac{kq-1}{3}$$$$\Rightarrow 3kp-3=2kq-2$$$$\Rightarrow (3p-2q)k=1$$$$\Rightarrow 3p-2q=\frac{1}{k}\ne integersincek\ne 1$$$$but3p-2qisinteger.$$$$\Rightarrow contradiction!\Rightarrow k=1$$

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