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Conjectured number of Fibonacci numbers with exactly n 0-bits in their binary representation.
+10
6
3, 5, 2, 4, 3, 1, 2, 3, 5, 1, 3, 5, 3, 3, 2, 1, 6, 1, 2, 3, 3, 2, 0, 3, 5, 4, 4, 3, 4, 2, 1, 2, 6, 1, 2, 2, 5, 4, 3, 5, 2, 2, 2, 1, 2, 2, 2, 5, 6, 3, 2, 2, 3, 1, 5, 1, 1, 0, 8, 4, 3, 3, 3, 3, 5, 4, 4, 2, 2, 2, 2, 3, 2, 6, 3, 0, 0, 2, 5, 5, 1, 6, 5, 0, 3, 5, 1
MATHEMATICA
f = Fibonacci[Range[0, 100]]; Table[Length[Select[f, Count[IntegerDigits[#, 2], 0] == n &]], {n, 0, 20}]
Irregular triangle of conjectured Fibonacci numbers with exactly n 0-bits in their binary representation.
+10
6
1, 1, 3, 0, 2, 5, 13, 55, 21, 987, 8, 89, 233, 377, 34, 1597, 28657, 6765, 144, 610, 17711, 196418, 514229, 2584, 4181, 10946, 121393, 317811, 3524578, 46368, 1346269, 1836311903, 75025, 5702887, 24157817, 102334155, 165580141, 832040, 14930352, 701408733
EXAMPLE
The irregular triangle begins
{1, 1, 3},
{0, 2, 5, 13, 55},
{21, 987},
{8, 89, 233, 377},
{34, 1597, 28657},
{6765},
{144, 610},
{17711, 196418, 514229},
{2584, 4181, 10946, 121393, 317811},
{3524578}, {46368, 1346269, 1836311903},
{75025, 5702887, 24157817, 102334155, 165580141},
{832040, 14930352, 701408733}
MATHEMATICA
f = Fibonacci[Range[0, 1000]]; Table[Select[f, Count[IntegerDigits[#, 2], 0] == n &], {n, 0, 20}]
Irregular triangle read by rows: row n lists the Fibonacci numbers with exactly n 1's in their binary representation.
+10
5
0, 1, 1, 2, 8, 3, 5, 34, 144, 13, 21
COMMENTS
Besides those listed in Example section, there are no additional terms with small number of 1's in the first 10^12 Fibonacci numbers. In particular, if A000120(Fibonacci(n)) < 100, then n <= 319 or n > 10^12. - Charles R Greathouse IV, Mar 06 2014 For the theorem about S-units that Noam Elkies quotes (in the MathOverflow link), see Chapter 1 of Storey-Tijdemann, 1986. - N. J. A. Sloane, Jan 28 2017
REFERENCES
T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge Tracts in Mathematics, 1986.
EXAMPLE
The irregular table begins
{0},
{1, 1, 2, 8},
{3, 5, 34, 144},
{13, 21, ...}.
It is conjectured that the previous (n=3) row is complete, and that the subsequent rows are:
{89, 610, 2584},
{55, 233, 4181},
{377, 10946, 46368, 75025},
{1597},
{987, 6765, 17711, 832040},
{121393, 2178309},
{39088169},
{28657, 196418, 317811, 1346269, 9227465},
{514229, 5702887, 14930352, 63245986, 4807526976},
{3524578, 2971215073}
...
MATHEMATICA
f = Fibonacci[Range[0, 100]]; Table[Select[f, Total[IntegerDigits[#, 2]] == n &], {n, 0, 20}]
PROG
(PARI) row(n)=my(k=-1, t); while(1, t=fibonacci(k++); if(hammingweight(t)==n, print1(t", "))) \\ Charles R Greathouse IV, Mar 04 2014
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