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Erdős-Mirsky Conjecture

The Erdős-Mirsky conjecture asserts that infinitely many positive integers n satisfy d(n)=d(n+1), where d(n) is the divisor function counting positive divisors (Erdős and Mirsky 1952). For example, d(2)=d(3)=2 and d(14)=d(15)=4.

Heath-Brown (1984) proved the conjecture. This is problem 946 in the Erdős problems collection (Bloom 2026).

See also

Divisor Function, Erdős Problems

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References

Bloom, T. F. "Erdős Problem 946." Accessed Oct. 26, 2026. https://www.erdosproblems.com/946.Erdős, P. and Mirsky, L. "The Distribution of Values of the Divisor Function d(n)." Proc. London Math. Soc. 2, 257-271, 1952. https://doi.org/10.1112/plms/s3-2.1.257.Heath-Brown, D. R. "The Divisor Function at Consecutive Integers." Mathematika 31, 141-149, 1984. https://doi.org/10.1112/S0025579300010743.

Cite this as:

Weisstein, Eric W. "Erdős-Mirsky Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Erdos-MirskyConjecture.html

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