# A positive Bézout's identity [duplicate]

Question: Is the following statement true?

Statement: Let $a, b \ge 1$ be coprime numbers. Then $\exists N \ge 0$ $\forall n \ge N$ $\exists u, v \ge 0$ with $n=au+bv$.
Let $N(a,b)$ be the smallest possible $N$. Then $N(a,b) = (a-1)(b-1)$.

## marked as duplicate by Bill Dubuque elementary-number-theory StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jul 8 '17 at 13:40

• Yes. If memory serves, $N-1$ is called the Frobenius number of $a$ and $b$. – Daniel Fischer Jul 8 '17 at 12:23