 TOPICS  # Golden Ratio Digits

The golden ratio has decimal expansion (OEIS A001622). It can be computed to digits of precision in 24 CPU-minutes on modern hardware and was computed to decimal digits by A. J. Yee on Jul. 8, 2010.

The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 2, 62, 158, 1216, 72618, 2905357, 7446157, 41398949, 1574998166, ... (OEIS A224844).

The digit sequence 0123456789 does not occur in the first digits of , but 9876543210 does, starting at position (E. Weisstein, Jul. 22, 2013).

Phi-primes, i.e., -constant primes occur for 7, 13, 255, 280, 97241, ... (OEIS A064119) 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, 20, 6, 12, 23, 2, 11, 4, 8, 232, ... (OEIS A088577).

Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 5, 55, 515, 0092, 67799, 290503, ... (OEIS A000000), which end at digits 23, 770, 5819, 93910, 1154766, 13192647, ... (OEIS A000000).

It is not known if is normal, 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 1 11 100 1020 9986 99805 1001143 10003332 100007840 1000031042 1 A000000 1 9 105 1062 9963 99993 1000118 10000255 99999864 999990982 2 A000000 0 11 116 994 9950 99529 1000776 10002116 100002106 1000005392 3 A000000 2 9 88 1039 10079 99833 999794 9999184 99979352 999978183 4 A000000 0 12 92 976 10041 100151 999367 9998797 99995481 999952470 5 A000000 0 5 84 988 10016 100067 999725 9996151 99999934 1000032985 6 A000000 1 9 104 918 9975 100328 999455 9996149 100004208 1000014191 7 A000000 1 10 113 1025 9988 100160 1000852 9997524 100018237 1000023870 8 A000000 3 15 105 987 10008 100236 1000059 10005419 99995223 999976728 9 A000000 1 9 93 991 9994 99898 998711 10001073 99997755 999994157

Constant Digit Scanning, Constant Primes, Golden Ratio

## Explore with Wolfram|Alpha ## References

Sloane, N. J. A. Sequences A/M4046, A064119, A088577, and A224844 in "The On-Line Encyclopedia of Integer Sequences."Yee, A. J. "y-cruncher - A Multi-Threaded Pi-Program." http://www.numberworld.org/y-cruncher/.

## Cite this as:

Weisstein, Eric W. "Golden Ratio Digits." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GoldenRatioDigits.html