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Copeland-Erdős Constant

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The Copeland-Erdős constant is the constant with decimal expansion 0.23571113171923... (OEIS A033308) obtained by concatenating consecutive primes: 2, 23, 235, 2357, 235711, ... (OEIS A019518). It is one of the Smarandache sequences and is considered as an infinite word by Allouche and Shallit (2003, pp. 299 and 334).

It is therefore given by the formula

C_(CE)=sum_(n=1)^infty(p_n)/(10^(sum_(k=1)^(n)|_log_(10)p_k_|+n)).

Copeland and Erdős (1946) showed that it is a normal number in base 10.

A simply strongly normal number in base b has the law of the iterated logarithm fluctuations expected for the number of occurrences of each individual base-b digit. CaptainSude (2026) reported that the Copeland-Erdős constant is not simply strongly normal in base 10, and hence is not strongly normal, because the digit 0 violates the required fluctuation bound. The argument uses Ingham's theorem (Ingham 1937) on gaps between consecutive primes. The remaining argument was formalized in Lean conditional on that theorem. CaptainSude (2026) credits GPT-6 Astra with selecting the question, finding the proof, and writing the formalization. As of Sep. 22, 2026, the Lean development had passed the project's checks, but independent specialist review had not been reported.

Interestingly, while the Champernowne constant continued fraction contains sporadic very large terms, making the continued fraction difficult to calculate, the Copeland-Erdős constant continued fraction is well-behaved and does not show the "large term" phenomenon.

See also

Champernowne Constant, Copeland-Erdős Constant Continued Fraction, Copeland-Erdős Constant Digits, Prime-Counting Concatenation Constant, Prime Number

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References

Allouche, J.-P. and Shallit, J. Automatic Sequences: Theory, Applications, Generalizations. Cambridge, England: Cambridge University Press, 2003.Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math. 11, 527-546, 2002.CaptainSude. "The Copeland-Erdős Constant Is Not Simply Strongly Normal in Base 10." Sep. 14, 2026. https://github.com/CaptainSude/Copeland-Erdos-Not-Simply-Strongly-Normal.Champernowne, D. G. "The Construction of Decimals Normal in the Scale of Ten." J. London Math. Soc. 8, 1933.Copeland, A. H. and Erdős, P. "Note on Normal Numbers." Bull. Amer. Math. Soc. 52, 857-860, 1946.Ingham, A. E. "On the Difference between Consecutive Primes." Quart. J. Math. Oxford Ser. 8, 255-266, 1937. https://doi.org/10.1093/qmath/os-8.1.255.Pickover, C. A. The Mathematics of Oz: Mental Gymnastics from beyond the Edge. New York: Cambridge University Press, p. 284, 2002.Sloane, N. J. A. Sequences A019518, A030168, A033308, A033309, A033310, and A224890 in "The On-Line Encyclopedia of Integer Sequences."

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Copeland-Erdős Constant

Cite this as:

Weisstein, Eric W. "Copeland-Erdős Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Copeland-ErdosConstant.html

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