9.34 Integer¶
FriCAS provides many operations for manipulating arbitrary precision integers. In this section we will show some of those that come from Integer itself plus some that are implemented in other packages. More examples of using integers are in the following sections: ugIntroNumbersPage in section ugIntroNumbersNumber IntegerNumberTheoryFunctionsXmpPage , DecimalExpansionXmpPage , BinaryExpansionXmpPage , HexadecimalExpansionXmpPage , and RadixExpansionXmpPage .
9.34.1 Basic Functions¶
The size of an integer in FriCAS is only limited by the amount of computer storage you have available. The usual arithmetic operations are available.
2^(5678 - 4856 + 2 * 17)
| 480481077043500814718154092512592439123952613987168226347385561008808420007630829308634252709141208374307457227821149607627692202643343568752733498024953930242542523045817764949544214392905306388478705146745768073877141698859815495632935288783334250628775936 |
Type: PositiveInteger
There are a number of ways of working with the sign of an integer. Let’s use this x as an example.
x := -101
| -101 |
Type: Integer
First of all, there is the absolute value function.
abs(x)
| 101 |
Type: PositiveInteger
The signsignInteger operation returns -1 if its argument is negative, 0 if zero and 1 if positive.
sign(x)
| -1 |
Type: Integer
You can determine if an integer is negative in several other ways.
x < 0
| true |
Type: Boolean
x <= -1
| true |
Type: Boolean
negative?(x)
| true |
Type: Boolean
Similarly, you can find out if it is positive.
x > 0
| false |
Type: Boolean
x >= 1
| false |
Type: Boolean
positive?(x)
| false |
Type: Boolean
This is the recommended way of determining whether an integer is zero.
zero?(x)
| false |
Type: Boolean
mathematical object for equality with zero. This is usually more efficient that using = (think of matrices: it is easier to tell if a matrix is zero by just checking term by term than constructing another zero matrix and comparing the two matrices term by term) and also avoids the problem that = is usually used for creating equations.
This is the recommended way of determining whether an integer is equal to one.
one?(x)
| false |
Type: Boolean
This syntax is used to test equality using =. It says that you want a Boolean (true or false) answer rather than an equation.
(x = -101)@Boolean
| true |
Type: Boolean
The operations odd?odd?Integer and even?even?Integer determine whether an integer is odd or even, respectively. They each return a Boolean object.
odd?(x)
| true |
Type: Boolean
even?(x)
| false |
Type: Boolean
The operation gcdgcdInteger computes the greatest common divisor of two integers.
gcd(56788,43688)
| 4 |
Type: PositiveInteger
The operation lcmlcmInteger computes their least common multiple.
lcm(56788,43688)
| 620238536 |
Type: PositiveInteger
To determine the maximum of two integers, use maxmaxInteger.
max(678,567)
| 678 |
Type: PositiveInteger
To determine the minimum, use minminInteger.
min(678,567)
| 567 |
Type: PositiveInteger
The reduce operation is used to extend binary operations to more than two arguments. For example, you can use reduce to find the maximum integer in a list or compute the least common multiple of all integers in the list.
reduce(max,[2,45,-89,78,100,-45])
| 100 |
Type: PositiveInteger
reduce(min,[2,45,-89,78,100,-45])
| -89 |
Type: Integer
reduce(gcd,[2,45,-89,78,100,-45])
| 1 |
Type: PositiveInteger
reduce(lcm,[2,45,-89,78,100,-45])
| 1041300 |
Type: PositiveInteger
The infix operator / is not used to compute the quotient of integers. Rather, it is used to create rational numbers as described in FractionXmpPage .
13 / 4
| 134 |
Type: Fraction Integer
The infix operation quoquoInteger computes the integer quotient.
13 quo 4
| 3 |
Type: PositiveInteger
The infix operation remremInteger computes the integer remainder.
13 rem 4
| 1 |
Type: PositiveInteger
One integer is evenly divisible by another if the remainder is zero. The operation exquoexquoInteger can also be used. See ugTypesUnionsPage in Section ugTypesUnionsNumber for an example.
zero?(167604736446952 rem 2003644)
| true |
Type: Boolean
The operation dividedivideInteger returns a record of the quotient and remainder and thus is more efficient when both are needed.
d := divide(13,4)
| [quotient=3,remainder=1] |
Type: Record(quotient: Integer,remainder: Integer)
d.quotient
| 3 |
Type: PositiveInteger
Records are discussed in detail in Section ugTypesRecords .
d.remainder
| 1 |
Type: PositiveInteger
9.34.2 Primes and Factorization¶
Use the operation factorfactorInteger to factor integers. It returns an object of type Factored Integer. See FactoredXmpPage for a discussion of the manipulation of factored objects.
factor 102400
| 21252 |
Type: Factored Integer
The operation prime?prime?Integer returns true or false depending on whether its argument is a prime.
prime? 7
| true |
Type: Boolean
prime? 8
| false |
Type: Boolean
The operation nextPrimenextPrimeIntegerPrimesPackage returns the least prime number greater than its argument.
nextPrime 100
| 101 |
Type: PositiveInteger
The operation prevPrimeprevPrimeIntegerPrimesPackage returns the greatest prime number less than its argument.
prevPrime 100
| 97 |
Type: PositiveInteger
To compute all primes between two integers (inclusively), use the operation primesprimesIntegerPrimesPackage.
primes(100,175)
| [173,167,163,157,151,149,139,137,131,127,113,109,107,103,101] |
Type: List Integer
You might sometimes want to see the factorization of an integer when it is considered a Gaussian integer. See ComplexXmpPage for more details.
factor(2 :: Complex Integer)
| -i(1+i)2 |
Type: Factored Complex Integer
9.34.3 Some Number Theoretic Functions¶
FriCAS provides several number theoretic operations for integers. More examples are in IntegerNumberTheoryFunctionsXmpPage .
The operation fibonaccifibonacciIntegerNumberTheoryFunctions computes the Fibonacci numbers. The algorithm has running time for argument n.
[fibonacci(k) for k in 0..]
| [0,1,1,2,3,5,8,13,21,34,…] |
Type: Stream Integer
The operation legendrelegendreIntegerNumberTheoryFunctions computes the Legendre symbol for its two integer arguments where the second one is prime. If you know the second argument to be prime, use jacobijacobiIntegerNumberTheoryFunctions instead where no check is made.
[legendre(i,11) for i in 0..10]
| [0,1,-1,1,1,1,-1,-1,-1,1,-1] |
Type: List Integer
The operation jacobijacobiIntegerNumberTheoryFunctions computes the Jacobi symbol for its two integer arguments. By convention, 0 is returned if the greatest common divisor of the numerator and denominator is not 1.
[jacobi(i,15) for i in 0..9]
| [0,1,1,0,1,0,0,-1,1,0] |
Type: List Integer
The operation eulerPhieulerPhiIntegerNumberTheoryFunctions computes the values of Euler’s -function where equals the number of positive integers less than or equal to n that are relatively prime to the positive integer n.
[eulerPhi i for i in 1..]
| [1,1,2,2,4,2,6,4,6,4,…] |
Type: Stream Integer
The operation moebiusMumoebiusMuIntegerNumberTheoryFunctions computes the Möbius λ function.
[moebiusMu i for i in 1..]
| [1,-1,-1,0,-1,1,-1,0,0,1,…] |
Type: Stream Integer
Although they have somewhat limited utility, FriCAS provides Roman numerals.
a := roman(78)
| LXXVIII |
Type: RomanNumeral
b := roman(87)
| LXXXVII |
Type: RomanNumeral
a + b
| CLXV |
Type: RomanNumeral
a * b
| MMMMMMDCCLXXXVI |
Type: RomanNumeral
b rem a
| IX |
Type: RomanNumeral