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A positive integer is prime if it is divisible only by itself and 1. For technical reasons, 1 is not considered prime; the smallest prime is 2.
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Prime[n] returns the nth prime. RandomPrime[n] returns a pseudorandom prime number between 2 and n. RandomPrime[{m, n}] returns a pseudorandom prime number between m and n. RandomPrime[ { m, n } , k] returns a list of k pseudorandom primes, each between m and n.
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EXAMPLE 10 Find the 7th prime.
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Prime[7] 17 RandomPrime[{7, 47}] 29 RandomPrime[{7, 47}, 10] {31, 29, 41, 47, 43, 13, 31, 17, 37, 7}
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The Fibonacci numbers are defined by
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f1 = 1, f2 = 1, fn = fn 2 + fn 1 n 3
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Thus, the first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21,
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Fibonacci[n] returns the nth Fibonacci number.
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Fibonacci[7] 13
There are three Mathematica functions that convert real numbers to nearby integers.
Round[x] returns the integer closest to x. If x lies exactly between two integers (e.g., 5.5), Round returns the nearest even integer. Floor[x] returns the greatest integer which does not exceed x. This is sometimes known as the greatest integer function and is represented in many textbooks by x . Ceiling[x] returns the smallest integer not less than x. Many textbooks represent this by x .
EXAMPLE 12
Round[5.75] 6 Floor[5.75] 5 Ceiling[5.75] 6
A decimal number can be broken up into two parts, the integer portion (number to the left of the decimal point) and the fractional portion.
IntegerPart[x] gives the integer portion of x (decimal point excluded). FractionalPart[x] gives the fractional portion of x (decimal point included).
Observe that IntegerPart[x]+ FractionalPart[x]= x.
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EXAMPLE 13
IntegerPart[4.67] 4 FractionalPart[4.67] 0.67 IntegerPart[4.67] + FractionalPart[4.67] 4.67
If m and n are positive integers, there exist unique integers q and r such that m = qn + r with 0 r<n
This result is known as the Division Algorithm. q is called the quotient and r is the remainder. The Mathematica functions Quotient and Mod return the quotient and remainder, respectively.
Quotient[m, n] returns the quotient when m is divided by n. Mod[m, n] returns the remainder when m is divided by n.
EXAMPLE 14
Quotient[17, 3] 5 Mod[17, 3] 2
Suppose a and b are two integers. If there exists an integer, k, such that a = kb, we say that b divides a. Alternatively, a is a multiple of b. Let m and n be two integers. If b divides both m and n, we say that b is a common divisor of m and n. The largest common divisor of m and n is called their greatest common divisor (GCD). If a is a multiple of both m and n, we say a is a common multiple of m and n. The smallest common multiple of m and n is called their least common multiple (LCM).
GCD[m, n] returns the greatest common divisor of m and n. LCM[m, n] returns the least common multiple of m and n.
The functions GCD and LCM extend to more than two arguments.
EXAMPLE 15 Find the greatest common divisor and least common multiple of 24, 40, and 48.
GCD[24, 40, 48] 8 LCM[24, 40, 48] 240
The Fundamental Theorem of Arithmetic guarantees that every positive integer can be factored into primes in a unique way.
The function FactorInteger[n] gives the prime factors of n together with their respective exponents.
EXAMPLE 16
FactorInteger[2 381 400] {{2, 3} ,{3, 5}, {5, 2}, {7, 2}}
The prime factors of 2,381,400 are 2, 3, 5, and 7 with exponents, respectively, 3, 5, 2, 2. In other words, 2,381,400 = 23355272. The result of this operation produces a nested sequence of lists. (A list is a Mathematica object, enclosed within braces, { }, which will be discussed in detail in 3.)
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In order to estimate computational efficiency, it is useful to be able to determine how long an operation or sequence of operations takes to execute.
Timing[expression] evaluates expression, and returns a list of time used, in seconds, together with the result obtained.
Timing counts only the CPU time spent in the Mathematica kernel. It does not include overhead time spent in the front end.
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