Get ready Part 7

A composite (positive integer) is a product a b ab with a a and b b not necessarily distinct integers in { 2 , 3 , 4 , } \{2,3,4,\dots\} .Is every composite is expressible as x y + x z + y z + 1 xy+xz+yz+1 ,with x , y , z x,y,z positive integers?

No Yes

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1 solution

Otto Bretscher
Oct 28, 2018

If we write a = x + 1 a=x+1 and b = y + 1 b=y+1 then a b = ( x + 1 ) ( y + 1 ) = x y + x + y + 1 ab=(x+1)(y+1)=xy+x+y+1 . Now let z = 1 z=1 .

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