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Theodorus's Constant Digits

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Theodorus's constant sqrt(3) has decimal expansion

sqrt(3)=1.732050807...

(OEIS A002194). It was computed to 10^(10) decimal digits by E. Weisstein on Jul. 23, 2013.

The Earls sequence (starting position of n copies of the digit n) for e is given for n=1, 2, ... by 27, 215, 1651, 2279, 21640, 176497, 7728291, 77659477, 638679423, ... (OEIS A224874).

sqrt(3)-constant primes occur at 2, 3, 19, 111, 116, 641, 5411, 170657, ... (OEIS A119344) decimal digits.

The starting positions of the first occurrence of n=0, 1, 2, ... in the decimal expansion of sqrt(3) (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 sqrt(3) until all n-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 10^(10) digits of sqrt(3), but 0123456789 does, starting at positions 1104282392, 1879095207, 3037917993, ... (OEIS A000000) (E. Weisstein, Jul. 23, 2013).

It is not known if sqrt(3) is normal (Beyer et al. 1969, 1970ab), but the following table giving the counts of digits in the first 10^n terms shows that the decimal digits are very uniformly distributed up to at least 10^(10).

d\nOEIS1010010^310^410^510^610^710^810^910^(10)
03159510351012510023410001729995281999766381000006042
10797996100199958710015481000167099988551999978902
21810099498299981210002631000175199991487999982296
3199794598989981899894310000247100004464999998469
407849711007799897998647100013841000232031000009144
521393100910037100260999993999587999996674999982506
601010310271005210055899997699999311000201481000025094
72119899199219992110000591000265599987934999997927
8114125100299961000551000650100010421000171071000013674
9061081030100469985899974910000160999937941000005946

See also

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."

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Theodorus's Constant Digits

Cite this as:

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

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