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Pisano Period
In number theory, the ''n''th Pisano period, written as '(''n''), is the period with which the sequence of Fibonacci numbers taken modulo ''n'' repeats. Pisano periods are named after Leonardo Pisano, better known as Fibonacci. The existence of periodic functions in Fibonacci numbers was noted by Joseph Louis Lagrange in 1774. Definition The Fibonacci numbers are the numbers in the integer sequence: :0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, ... defined by the recurrence relation :F_0 = 0 :F_1 = 1 :F_i = F_ + F_. For any integer ''n'', the sequence of Fibonacci numbers ''Fi'' taken modulo ''n'' is periodic. The Pisano period, denoted '(''n''), is the length of the period of this sequence. For example, the sequence of Fibonacci numbers modulo 3 begins: :0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, ... This sequence has period 8, so '(3) = 8. Properties Parity Wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pisano Periods
Pisano may refer to: * Pisano, a native or inhabitant of Pisa, Italy * Pisano (surname), a list of people * Pisano, Piedmont, commune in the province of Novara, in northern Italy * Pisano period, in number theory * Pisanello (1380–1456), Italian artist sometimes erroneously called Vittore Pisano See also * * Pisa (other) * Pisana (other) * Pisani (other) * Pisanu, a surname {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integers Modulo N
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This can b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Of A Function
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equivalently, x is a solution to the equation f(x) = 0. A "zero" of a function is thus an input value that produces an output of 0. A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by f(x)=x^2-5x+6=(x-2)(x-3) has the two roots (or zeros) that are 2 and 3. f(2)=2^2-5\times 2+6= 0\textf(3)=3^2-5\times 3+6=0. If the function maps real numbers to real n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Order
In number theory, given a positive integer ''n'' and an integer ''a'' coprime to ''n'', the multiplicative order of ''a'' modulo ''n'' is the smallest positive integer ''k'' such that a^k\ \equiv\ 1 \pmod n. In other words, the multiplicative order of ''a'' modulo ''n'' is the order of ''a'' in the multiplicative group of the units in the ring of the integers modulo ''n''. The order of ''a'' modulo ''n'' is sometimes written as \operatorname_n(a). Example The powers of 4 modulo 7 are as follows: : \begin 4^0 &= 1 &=0 \times 7 + 1 &\equiv 1\pmod7 \\ 4^1 &= 4 &=0 \times 7 + 4 &\equiv 4\pmod7 \\ 4^2 &= 16 &=2 \times 7 + 2 &\equiv 2\pmod7 \\ 4^3 &= 64 &=9 \times 7 + 1 &\equiv 1\pmod7 \\ 4^4 &= 256 &=36 \times 7 + 4 &\equiv 4\pmod7 \\ 4^5 &= 1024 &=146 \times 7 + 2 &\equiv 2\pmod7 \\ \vdots\end The smallest positive integer ''k'' such that 4''k'' ≡ 1 (mod 7) is 3, so the order of 4 (mod 7) is 3. Properties Even without knowledge that we are working in the multiplicative ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binet's Formula
In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many writers begin the sequence with 0 and 1, although some authors start it from 1 and 1 and some (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the sequence begins : 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... The Fibonacci numbers were first described in Indian mathematics as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book . Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''Fibonacci Quarterly''. Appli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Definition An integer n is divisible by a nonzero integer m if there exists an integer k such that n=km. This is written as : m\mid n. This may be read as that m divides n, m is a divisor of n, m is a factor of n, or n is a multiple of m. If m does not divide n, then the notation is m\not\mid n. There are two conventions, distinguished by whether m is permitted to be zero: * With the convention without an additional constraint on m, m \mid 0 for every integer m. * With the convention that m be nonzero, m \mid 0 for every nonzero integer m. General Divisors can be negative as well as positive, although often the term is restricted to posi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wall–Sun–Sun Prime
In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known. Definition Let p be a prime number. When each term in the sequence of Fibonacci numbers F_n is reduced modulo p, the result is a periodic sequence. The (minimal) period length of this sequence is called the Pisano period and denoted \pi(p). Since F_0 = 0, it follows that ''p'' divides F_. A prime ''p'' such that ''p''2 divides F_ is called a Wall–Sun–Sun prime. Equivalent definitions If \alpha(m) denotes the rank of apparition modulo m (i.e., \alpha(m) is the smallest positive index such that m divides F_), then a Wall–Sun–Sun prime can be equivalently defined as a prime p such that p^2 divides F_. For a prime ''p'' ≠ 2, 5, the rank of apparition \alpha(p) is known to divide p - \left(\tfrac\right), where the Legendre symbol \textstyle\left(\frac\right) has the values :\left(\frac\right) = \beg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are lazy." In mathematics In mathematics, counterexamples are often used to prove the boundaries of possible theorems. By using counterexamples to show that certain conjectures are false, mathematical researchers can then avoid going down blind alleys and learn to modify conjectures to produce provable theorems. It is sometimes said that mathematical development consists primarily in finding (and proving) theorems and counterexamples. Rectangle example Suppose that a mathematician is studying geometry and shapes, and she wishes to prove certain theorems about them. She conjectures that "All re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Power
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, … . The prime powers are those positive integers that are divisible by exactly one prime number; in particular, the number 1 is not a prime power. Prime powers are also called primary numbers, as in the primary decomposition. Properties Algebraic properties Prime powers are powers of prime numbers. Every prime power (except powers of 2 greater than 4) has a primitive root; thus the multiplicative group of integers modulo ''p''''n'' (that is, the group of units of the ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
