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A prime divides if and only if p is congruent to ±1 modulo 5, and p divides + if and only if it is congruent to ±2 modulo 5. (For p = 5, F 5 = 5 so 5 divides F 5) . Fibonacci numbers that have a prime index p do not share any common divisors greater than 1 with the preceding Fibonacci numbers, due to the identity: [6]
That is, after two starting values, each number is the sum of the two preceding numbers. The Fibonacci sequence has been studied extensively and generalized in many ways, for example, by starting with other numbers than 0 and 1, by adding more than two numbers to generate the next number, or by adding objects other than numbers.
A Fibonacci prime is a Fibonacci number that is prime. The first few are: [46] 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ... Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many. [47] F kn is divisible by F n, so, apart from F 4 = 3, any Fibonacci prime must have a prime index.
where F n is the n th Fibonacci number. Such a sum is called the Zeckendorf representation of N. The Fibonacci coding of N can be derived from its Zeckendorf representation. For example, the Zeckendorf representation of 64 is 64 = 55 + 8 + 1. There are other ways of representing 64 as the sum of Fibonacci numbers 64 = 55 + 5 + 3 + 1 64 = 34 ...
The prime quadruplets are pairs of twin primes with only one odd number between them. The sum of the reciprocals of the numbers in prime quadruplets is approximately 0.8706. The sum of the reciprocals of the perfect powers (including duplicates) is 1. The sum of the reciprocals of the perfect powers (excluding duplicates) is approximately 0. ...
These indices are all themselves prime. As with the Fibonacci numbers, a Pell number P n can only be prime if n itself is prime, because if d is a divisor of n then P d is a divisor of P n. The only Pell numbers that are squares, cubes, or any higher power of an integer are 0, 1, and 169 = 13 2. [7]
Individual numbers in the Lucas sequence are known as Lucas numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences. The Lucas sequence has the same recursive relationship as the Fibonacci sequence, where each term is the sum of the two previous terms, but with different starting values. [1]
Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime (the product of two primes). [64] Also, any even integer greater than can be written as the sum of six primes. [65] The branch of number theory studying such questions is called additive number theory. [66]