What non-zero Fibonacci numbers are one less than a power of two? (That would make each of them consist of all 1's in binary.)

(In reply to re: A possible solution (spoiler) by Charlie)

Perhaps it is more accurate to say there are (at least) three Fibonacci numbers which are one less than a power of two.  As the Fibonacci sequence begins 0, 1, 1, 2, 3, 5, 8, 13...., it can be seen that 1 is repeated twice -- of which both 1's are one less than a power of two (2¹ - 1 = 1).  Therefore, I shall say there is 1, 1, and 3.  Other than these three numbers, examining the pattern of two's complement arithmetic, I do not believe there is another.  If I am mistaken, then I must assume there is an infinite number of these numbers (though they would very well be astronomically huge and distant apart from each other).

Posted by Dej Mar on 2007-03-23 08:01:18