I have equation $z = 3x + 5y$ where $x$, $y$ are integers 0 or greater. How can I prove that there is no two (different) consecutive integers $z_1, z_2$, greater than 2, that cannot be built using this equation. So for example using this equation I cant make 17 but I can 16 and 18 so I could find some $x$, $y$ to find a consecutive number to 17.
2026-03-31 03:29:45.1774927785
Prove that there is no two consecutive natural numbers in a sum
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If $z_1$ and $z_2=z_1+1$ are consecutive positive integers that cannot be represented in that form, then certainly neither of them is a multiple of $3$. Hence $z_1=3k+1$, $z_2=3k+2$ for some $k\ge1$. But then $z_2-5=3(k-1)$ is a multiple of $3$ and hence representable.