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# Euler-Mascheroni Constant Digits

(OEIS A001620) was calculated to 16 digits by Euler in 1781 and to 32 decimal places by Mascheroni (1790), although only the first 19 decimal places were correct. It was subsequently computed to 40 correct decimal placed by Soldner in 1809 and verified by Gauss and Nicolai in 1812 (Havil 2003, pp. 89-90). No quadratically converging algorithm for computing is known (Bailey 1988).

The following table summarizes some record computations.

 decimal digit date reference Oct. 1999 X. Gourdon and P. Demichel (Gourdon and Sebah) Dec. 8, 2006 Alexander J. Yee (Yee 2006; United Press International 2007) ? S. Kondo Mar. 13, 2009 A. Yee

The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 5, 139, 163, 10359, 86615, 193446, 236542, 6186099, 36151186, ... (OEIS A224826).

-constant primes occur at 1, 3, 40, 185, 1038, 22610, 179849, ... (A065815) decimal digits.

The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (excluding the initial 0 to the left of the decimal point) are 11, 5, 4, 14, 9, 1, 7, 2, 16, 10, ... (OEIS A229192).

Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 8, 18, 346, 2778, 84514, ... (OEIS A000000), which end at digits 16, 658, 6600, 91101, 1384372, ... (OEIS A000000).

It is not known if is normal, but the following table giving the counts of digits in the first terms shows that the decimal digits are very uniformly distributed up to at least .

 OEIS 10 100 0 A000000 0 11 111 1004 10065 100150 999853 10001768 99998397 1 A000000 1 6 95 1006 9974 100143 1000601 9996653 100002318 2 A000000 1 10 97 967 9821 99796 998927 9998112 99986624 3 A000000 0 9 108 976 9973 100194 1000766 9999460 99984204 4 A000000 1 10 90 1014 9870 99783 1001444 10007542 100011681 5 A000000 2 9 99 980 10200 100110 1002104 10001985 99996372 6 A000000 2 14 90 988 10103 100170 999530 9996871 100014127 7 A000000 2 13 116 1014 9877 99682 998692 9997487 99988819 8 A000000 0 7 81 1033 10114 100135 998534 9998182 100006202 9 A000000 1 11 113 1018 10003 99837 999549 10001940 100011256

Constant Digit Scanning, Constant Primes, Earls Sequence, Euler-Mascheroni Constant, Euler-Mascheroni Constant Continued Fraction

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## References

Bailey, D. H. "Numerical Results on the Transcendence of Constants Involving , , and Euler's Constant." Math. Comput. 50, 275-281, 1988.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.Mascheroni, L. Adnotationes ad calculum integralem Euleri, Vol. 1 and 2. Ticino, Italy, 1790 and 1792. Reprinted in Euler, L. Leonhardi Euleri Opera Omnia, Ser. 1, Vol. 12. Leipzig, Germany: Teubner, pp. 415-542, 1915.Sloane, N. J. A. Sequences A001620/M3755, A065815, A224826, and A229192 in "The On-Line Encyclopedia of Integer Sequences."Yee, A. J. "Euler's Constant-116 Million Digits on a Laptop: New World Record." 2006. http://www.numberworld.org/euler116m.html.Yee, A. J. "y-cruncher - A Multi-Threaded Pi-Program." http://www.numberworld.org/y-cruncher/.United Press International. "Student at Northwestern Breaks Math Record." Apr. 9, 2007. http://www.upi.com/NewsTrack/Quirks/2007/04/09/student_at_northwestern_breaks_math_record/.

## Cite this as:

Weisstein, Eric W. "Euler-Mascheroni Constant Digits." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Euler-MascheroniConstantDigits.html