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Every positive integer N can be written uniquely as = = where the integer coefficients 0 ≤ b n ≤ a n and if b n = a n then b n−1 = 0.. If α is the golden ratio, then all the partial quotients a n are equal to 1, the denominators q n are the Fibonacci numbers and we recover Zeckendorf's theorem on the Fibonacci representation of positive integers as a sum of distinct non-consecutive ...
Lemma: The sum of any non-empty set of distinct, non-consecutive Fibonacci numbers whose largest member is F j is strictly less than the next larger Fibonacci number F j + 1 . The lemma can be proven by induction on j .
The condition of having no two consecutive ones, used in binary to define the fibbinary numbers, is the same condition used in the Zeckendorf representation of any number as a sum of non-consecutive Fibonacci numbers.
This is a list of articles about prime numbers. A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes.
Such a number is algebraic and can be expressed as the sum of a rational number and the square root of a rational number. Constructible number: A number representing a length that can be constructed using a compass and straightedge. Constructible numbers form a subfield of the field of algebraic numbers, and include the quadratic surds.
The even perfect numbers are not trapezoidal numbers; that is, they cannot be represented as the difference of two positive non-consecutive triangular numbers. There are only three types of non-trapezoidal numbers: even perfect numbers, powers of two, and the numbers of the form (+) formed as the product of a Fermat prime + with a power of two ...
A number that has the same number of digits as the number of digits in its prime factorization, including exponents but excluding exponents equal to 1. A046758: Extravagant numbers: 4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30, 33, 34, 36, 38, ... A number that has fewer digits than the number of digits in its prime factorization (including ...
Again, for example, if we begin with the number 42, this time as simply a positive integer, we have its binary representation 101010. This decodes to 2 0 · 3 1 · 5 0 · 7 1 · 11 0 · 13 1 = 3 × 7 × 13 = 273. Thus binary encoding of squarefree numbers describes a bijection between the nonnegative integers and the set of positive squarefree ...