There are Infinitely Many Carmichael Numbers
Theorem
There are infinitely many Carmichael numbers.
Proof
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Historical Note
When Robert Daniel Carmichael first identified in $1909$ 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 $1992$, when William Robert Alford, Andrew Granville and Carl Pomerance finally proved it.
Some sources give this date as $1994$, but that was just the date when the article they wrote on the subject was published.
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): $561$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): pseudoprime
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): pseudoprime