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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)
0A0000003159510351012510023410001729995281999766381000006042
1A0000000797996100199958710015481000167099988551999978902
2A0000001810099498299981210002631000175199991487999982296
3A000000199794598989981899894310000247100004464999998469
4A00000007849711007799897998647100013841000232031000009144
5A00000021393100910037100260999993999587999996674999982506
6A00000001010310271005210055899997699999311000201481000025094
7A0000002119899199219992110000591000265599987934999997927
8A000000114125100299961000551000650100010421000171071000013674
9A000000061081030100469985899974910000160999937941000005946

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."

Cite this as:

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

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