Erdős Conjecture on Carmichael Numbers

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

A Carmichael number is a composite positive integer $n$ for which $a^{n-1}\equiv1\pmod n$ whenever $\gcd(a,n)=1$. Let $C(x)$ count these integers below $x$. The asserted limit is $\log C(x)/\log x\to1$, equivalently $C(x)=x^{1-o(1)}$; this does not assert a positive natural density. [S1] [S2]

Statement recorded in the supplied source.

Let $C(x)$ be the number of Carmichael numbers less than $x$. Does $(\ln C(x))/\ln x$ tend to 1 as $x$ tends to infinity? [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Raise the unconditional lower exponent from just above 1/3 to 1-o(1), matching the logarithmic limit asserted by Erdős. [S1]

Short English statement

Do Carmichael numbers have a counting function with exponent tending to one, despite possibly forming a density-zero set?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 1866 (GUY-A13), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 Glyn Harman, On the number of Carmichael numbers up to x, Bulletin of the London Mathematical Society 37 (2005), 641-650. https://doi.org/10.1112/S0024609305004686. Bibliographic information from the supplied record.

• S3 Thomas Wright, A conditional density for Carmichael numbers, Bulletin of the Australian Mathematical Society 101 (2020), 379-388. https://doi.org/10.1017/S000497271900145X. Bibliographic information from the supplied record.

• S4 Erdős Problems, Problem 1057, Carmichael-number counting function. https://www.erdosproblems.com/1057. Bibliographic information from the supplied record.