TOPICS
Search

Pythagoras's Constant Digits

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

Pythagoras's constant sqrt(2) has decimal expansion

sqrt(2)=1.4142135623...

(OEIS A000129), It was computed to 2000000000050 decimal digits by A. J. Yee on Feb. 9, 2012.

The Earls sequence (starting position of n copies of the digit n) for e is given for n=1, 2, ... by 2, 114, 1481, 3308, 72459, 226697, 969836, 119555442, 2971094743, ... (OEIS A224871).

sqrt(2)-constant primes occur at 55, 97, 225, 11260, 11540, ... (OEIS A115377) decimal digits.

The starting positions of the first occurrence of n=0, 1, 2, ... in the decimal expansion of sqrt(2) (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 ln10 until all n-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 10^(10) digits of e, but 0123456789 does, starting at positions 864106288, 6458611884, 7311432557, ... (OEIS A000000) (E. Weisstein, Jul. 22, 2013).

It is not known if sqrt(2) 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)
001010895299599981499989710002237100010228999996989
1279810051010698924100011410000179999983811000042849
228109100498761004361000208999809199995645999987069
321182980100581001919996741000417899995415999984900
4291001016101001000241000126100000541000127251000008724
5171041001100021001559993589998344100002636999970045
61109010329939998861001246100016651000126831000007824
701810496410008100008999359999864699980315999986743
80121131027100071004419994529996550999951201000025363
908921019994510012110005661000005699996852999989494

See also

Constant Digit Scanning, Constant Primes, Pythagoras's Constant

Explore with Wolfram|Alpha

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." https://www.numberworld.org/y-cruncher/#Records.

Referenced on Wolfram|Alpha

Pythagoras's Constant Digits

Cite this as:

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

Subject classifications