Largest Number not Expressible as Sum of Multiples of 23 and 28

Theorem

The largest integer $n$ that cannot be expressed in the form:

$n = 23 x + 28 y$

for $x, y \in \Z_{>0}$ is $593$.

Proof

By Largest Number not Expressible as Sum of Multiples of Coprime Integers, the largest such number is:

$\paren {23 - 1} \times \paren {28 - 1} - 1 = 593$

This theorem requires a proof. In particular: This is a specific "historical" example of a general result which I read in Polya and Szego some time back, which still needs to be added into $\mathsf{Pr} \infty \mathsf{fWiki}$. All in due course. I may already have it on here, I misremember. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Historical Note

Wilhelm Fliess thought he had stumbled upon a remarkable fact about the numbers $23$ and $28$: that combinations of them could be used to explain practically every earthly phenomenon.

From this "discovery" be built the pseudoscience of biorhythms.

However, it needs to be noted that all but a finite set of positive integers can be represented in the form $23 x + 28 y$ for positive integers $x$ and $y$.

The number $593$ is the largest which cannot.

Every pair of coprime integers has the same property as $23$ and $28$.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $593$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $593$