There are Infinitely Many Carmichael Numbers

Theorem

There are infinitely many Carmichael numbers.

Proof

This theorem requires a proof. 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

When Robert Daniel Carmichael first identified in $1910$ the existence of what are now called Carmichael numbers, he expressed his belief that there were infinitely many.

This was the general (although unproven) belief in the mathematical community, up until $1994$, when W.R. Alford, Andrew Granville and Carl Pomerance finally proved it.

However, this information took some time to be widely disseminated, and in $1997$, David Wells was still reporting in his Curious and Interesting Numbers, 2nd ed. that:

It is widely believed, but not proved, that there are an infinite number of Carmichael numbers, but they are rare.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $561$
• 1994: W.R. Alford, Andrew Granville and Carl Pomerance: There are infinitely many Carmichael numbers (Ann. Math. Vol. 139, no. 3: pp. 703 – 722)  www.jstor.org/stable/2118576
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $561$