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# Pythagoras's Constant Digits

Pythagoras's constant has decimal expansion

(OEIS A000129), It was computed to decimal digits by A. J. Yee on Feb. 9, 2012.

The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 2, 114, 1481, 3308, 72459, 226697, 969836, 119555442, 2971094743, ... (OEIS A224871).

-constant primes occur at 55, 97, 225, 11260, 11540, ... (OEIS A115377) decimal digits.

The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (including the initial 1 and counting it as the first digit) are 14, 1, 5, 7, 2, 8, 9, 12, 19, ... (OEIS A229199).

Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 8, 81, 748, 8505, 30103, 489568, ... (OEIS A000000), which end at digits 19, 420, 8326, 94388, 1256460, 13043524, ... (OEIS A000000).

The digit sequence 9876543210 does not occur in the first digits of , but 0123456789 does, starting at positions 864106288, 6458611884, 7311432557, ... (OEIS A000000) (E. Weisstein, Jul. 22, 2013).

It is not known if is normal (Beyer et al. 1969, 1970ab), 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 10 108 952 9959 99814 999897 10002237 100010228 999996989 1 A000000 2 7 98 1005 10106 98924 1000114 10000179 99998381 1000042849 2 A000000 2 8 109 1004 9876 100436 1000208 9998091 99995645 999987069 3 A000000 2 11 82 980 10058 100191 999674 10004178 99995415 999984900 4 A000000 2 9 100 1016 10100 100024 1000126 10000054 100012725 1000008724 5 A000000 1 7 104 1001 10002 100155 999358 9998344 100002636 999970045 6 A000000 1 10 90 1032 9939 99886 1001246 10001665 100012683 1000007824 7 A000000 0 18 104 964 10008 100008 999359 9998646 99980315 999986743 8 A000000 0 12 113 1027 10007 100441 999452 9996550 99995120 1000025363 9 A000000 0 8 92 1019 9945 100121 1000566 10000056 99996852 999989494

Constant Digit Scanning, Constant Primes, Pythagoras's Constant

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## References

Beyer, W. A.; Metropolis, N.; and Neergaard, J. R. "Square Roots of Integers 2 to 15 in Various Bases 2 to 10: 88062 Binary Digits or Equivalent." Math. Comput. 23, 679, 1969.Beyer, W. A.; Metropolis, N.; and Neergaard, J. R. "Statistical Study of Digits of Some Square Roots of Integers in Various Bases." Math. Comput. 24, 455-473, 1970a.Beyer, W. A.; Metropolis, N.; and Neergaard, J. R. "The Generalized Serial Test Applied to Expansions of Some Irrational Square Roots in Various Bases." Math. Comput. 24, 745-747, 1970b.Sloane, N. J. A. Sequences A000129/M1314, A115377, A224871, and A229199 in "The On-Line Encyclopedia of Integer Sequences."Yee, A. J. "y-cruncher - A Multi-Threaded Pi-Program." http://www.numberworld.org/y-cruncher/#Records.

## Cite this as:

Weisstein, Eric W. "Pythagoras's Constant Digits." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PythagorassConstantDigits.html