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)
0A00000001010895299599981499989710002237100010228999996989
1A000000279810051010698924100011410000179999983811000042849
2A00000028109100498761004361000208999809199995645999987069
3A00000021182980100581001919996741000417899995415999984900
4A000000291001016101001000241000126100000541000127251000008724
5A000000171041001100021001559993589998344100002636999970045
6A0000001109010329939998861001246100016651000126831000007824
7A00000001810496410008100008999359999864699980315999986743
8A0000000121131027100071004419994529996550999951201000025363
9A00000008921019994510012110005661000005699996852999989494

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." 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

Subject classifications