 TOPICS # Theodorus's Constant Digits

Theodorus's constant has decimal expansion (OEIS A002194). It was computed to decimal digits by E. Weisstein on Jul. 23, 2013.

The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 27, 215, 1651, 2279, 21640, 176497, 7728291, 77659477, 638679423, ... (OEIS A224874). -constant primes occur at 2, 3, 19, 111, 116, 641, 5411, 170657, ... (OEIS A119344) 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 5, 1, 4, 3, 23, 6, 12, 2, 8, 18, ... (OEIS A229200).

Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 4, 91, 184, 5566, 86134, 35343, ... (OEIS A000000), which end at digits 23, 378, 7862, 77437, 1237533, 16362668, ... (OEIS A000000).

The digit sequence 9876543210 does not occur in the first digits of , but 0123456789 does, starting at positions 1104282392, 1879095207, 3037917993, ... (OEIS A000000) (E. Weisstein, Jul. 23, 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 3 15 95 1035 10125 100234 1000172 9995281 99976638 1000006042 1 A000000 0 7 97 996 10019 99587 1001548 10001670 99988551 999978902 2 A000000 1 8 100 994 9829 99812 1000263 10001751 99991487 999982296 3 A000000 1 9 97 945 9898 99818 998943 10000247 100004464 999998469 4 A000000 0 7 84 971 10077 99897 998647 10001384 100023203 1000009144 5 A000000 2 13 93 1009 10037 100260 999993 9995879 99996674 999982506 6 A000000 0 10 103 1027 10052 100558 999976 9999931 100020148 1000025094 7 A000000 2 11 98 991 9921 99921 1000059 10002655 99987934 999997927 8 A000000 1 14 125 1002 9996 100055 1000650 10001042 100017107 1000013674 9 A000000 0 6 108 1030 10046 99858 999749 10000160 99993794 1000005946

Constant Digit Scanning, Constant Primes, Earls Sequence, Theodorus'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 A002194/M4326, A119344, A224874, A229200 in "The On-Line Encyclopedia of Integer Sequences."

## Cite this as:

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