既然有特例,就无法证明!
如1、2^3+3^2=17是素数
2、当n=3k+1,nⁿ⁺¹+(n+1)ⁿ=(3k+1)^(3k+2)+(3k+2)^(3k+1)
模3知:3/1+2^(3k+1)=1+(-1)^(3k+1),因此只有n=3k+1为奇数,(3k+1)^(3k+2)+(3k+2)^(3k+1)>3,则(3k+1)^(3k+2)+(3k+2)^(3k+1)为合数!
3、当n=5k+1,nⁿ⁺¹+(n+1)ⁿ=(5k+1)^(5k+2)+(5k+2)^(5k+1)
模5知:1+2^(5k+1)=1+2^(k+1),当k+1=2a,a为奇数,(5k+1)^(5k+2)+(5k+2)^(5k+1)为合数,此时n=5k+1=5(2a-1)+1=10a-4