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Erdős Primitive Set Conjecture

The Erdős primitive set conjecture asserts that for any primitive sequence A of integers greater than 1,

sum_(a in A)1/(alna)<=sum_(p prime)1/(plnp).

Here no member of A divides another member, and ln is the natural logarithm. Thus the prime numbers maximize this weighted sum over all such sets (Lichtman 2023).

Erdős (1935) proved a uniform bound for these sums. Lichtman (2023) proved the sharper conjecture that the primes attain the maximum. Excluding 1 is essential because its logarithm is zero.

This is problem 164 in the Erdős problems collection (Bloom 2026).

See also

Erdős Problems, Primitive Sequence

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References

Bloom, T. F. "Erdős Problem 164." Accessed Oct. 26, 2026. https://www.erdosproblems.com/164.Erdős, P. "Note on Sequences of Integers No One of Which Is Divisible by Any Other." J. London Math. Soc. 10, 126-128, 1935. https://doi.org/10.1112/jlms/s1-10.1.126.Lichtman, J. D. "A Proof of the Erdős Primitive Set Conjecture." Forum Math. Pi 11, e18, 2023. https://doi.org/10.1017/fmp.2023.16.

Cite this as:

Weisstein, Eric W. "Erdős Primitive Set Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ErdosPrimitiveSetConjecture.html

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