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Turtle graphics are often associated with the Logo programming language. [2] Seymour Papert added support for turtle graphics to Logo in the late 1960s to support his version of the turtle robot, a simple robot controlled from the user's workstation that is designed to carry out the drawing functions assigned to it using a small retractable pen set into or attached to the robot's body.
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]
Examining divisibility by 5 as well, remainders upon division by 15 repeat with pattern 1, 11, 14, 10, 14, 11, 1, 14, 5, 4, 11, 11, 4, 5, 14 for the first polynomial, and with pattern 5, 0, 3, 14, 3, 0, 5, 3, 9, 8, 0, 0, 8, 9, 3 for the second, implying that only three out of 15 values in the second sequence are potentially prime (being ...
Thus the first term to appear between 1 / 3 and 2 / 5 is 3 / 8 , which appears in F 8. The total number of Farey neighbour pairs in F n is 2| F n | − 3. The Stern–Brocot tree is a data structure showing how the sequence is built up from 0 (= 0 / 1 ) and 1 (= 1 / 1 ), by taking successive mediants.
f p+1 ≡ 0 (mod p), where f k is the k-th Fibonacci number. The first condition is the Fermat primality test using base 2. In general, if p ≡ a (mod x 2 +4), where a is a quadratic non-residue (mod x 2 +4) then p should be prime if the following conditions hold: 2 p−1 ≡ 1 (mod p), f(1) p+1 ≡ 0 (mod p), f(x) k is the k-th Fibonacci ...
The prime number theorem then states that x / log x is a good approximation to π(x) (where log here means the natural logarithm), in the sense that the limit of the quotient of the two functions π(x) and x / log x as x increases without bound is 1:
As of 2024, it is known that F n is composite for 5 ≤ n ≤ 32, although of these, complete factorizations of F n are known only for 0 ≤ n ≤ 11, and there are no known prime factors for n = 20 and n = 24. [5] The largest Fermat number known to be composite is F 18233954, and its prime factor 7 × 2 18233956 + 1 was discovered in October 2020.
Certain finite simple groups can be recognized by the degrees of the vertices in their prime graphs. [3] The connected components of a prime graph have diameter at most five, and at most three for solvable groups. [4] When a prime graph is a tree, it has at most eight vertices, and at most four for solvable groups. [5]