Number Theory¶

General Functions¶

simula.ntheory.functions.Abs(f)¶

Returns the absolute value of f i. e |f|.

EXAMPLES:

simula :  Abs(2-pi)
-2 + pi
simula : Abs(x)
Abs(x)

class simula.ntheory.functions.Integer(i)¶

class simula.ntheory.functions.IntegerFactorization(i)¶: Representation of the prime decomposition of an integer.

simula.ntheory.functions.Max(*args)¶

Returns the maximum of args.

EXAMPLES:

simula : Max(4, 8, 9, 5)
9
simula : Max([4, 8, 9, 5, 20])
20

simula.ntheory.functions.Min(*args)¶

Returns the minimun of args.

EXAMPLES:

simula : Min(4, 8, 9, 5)
4
simula : Min([4, 8, 9, 2, 5, 20])
2

simula.ntheory.functions.Mod(g, f)¶: Represents a modulo operation on symbolic expressions i. e g modulo f.

simula.ntheory.functions.N(x, *, precision=15)¶

Returns a numerical approximation of x for a given precision precision (default : 15).

EXAMPLES :

simula : numerical_approx(pi)
3.141592653589793
simula : numerical_approx(pi, precision=20)
3.14159265358979323846

simula.ntheory.functions.beta(a, b)¶

Returns Beta(a, b).

EXAMPLES:

simula :  beta(3, 1)
1/3
simula : numerical_approx(beta(2, 2))
0.1666666666666667

simula.ntheory.functions.binomial(n, k)¶: Returns the binomial coefficient \(\dfrac{n!}{(n-k)! \times k! }\).

simula.ntheory.functions.ceil(a)¶: Returns the smallest integer value not less than a.

simula.ntheory.functions.denominator(expr)¶

Returns the denominator of expr.

EXAMPLES:

simula : f = 2/(sqrt(3)-1); f
2
simula : denominator(f)
-1 + sqrt(3)
simula : denominator(3pi/5)
5

simula.ntheory.functions.ellipsis_range(a, b, c=None)¶

Returns [a, b, a + (b-a), ..., c] if c is not None otherwise [a, a+1, ..., b].

EXAMPLES:

simula : ellipsis_range(2, 5)
[2, 3, 4, 5]
simula : ellipsis_range(2, 3, 10)
[2, 3, 4, 5, 6, 7, 8, 9, 10]
simula : ellipsis_range(1, 1.5, 5)
[1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0]

simula.ntheory.functions.euler_phi(n)¶

Returns the image of the Euler totient function at n.

EXAMPLES:

simula : euler_phi(3)
2
simula : euler_phi(10)
4
simula : euler_phi(15)
8

simula.ntheory.functions.evalf(x, *, precision=15)¶

Returns a numerical approximation of x for a given precision precision (default : 15).

EXAMPLES :

simula : numerical_approx(pi)
3.141592653589793
simula : numerical_approx(pi, precision=20)
3.14159265358979323846

simula.ntheory.functions.factorial(n)¶

Returns the factorial of n.

Parameters: n – a nonnegative integer

EXAMPLES:

simula : factorial(4)
24
simula : factorial(5)
120

simula.ntheory.functions.floor(a)¶: Returns the largest integer value not greater than a.

simula.ntheory.functions.fraction(a, b=1)¶

Returns a/b.

EXAMPLES:

simula : fraction(3,5)
3/5

simula.ntheory.functions.gamma(n)¶

Returns gamma(n).

EXAMPLES:

simula : gamma(2)
1
simula : gamma(6)
120
simula : gamma(2.5)
1.32934038817914

simula.ntheory.functions.integer_decomposition(n)¶: Returns the prime decomposition of the integer n.

simula.ntheory.functions.inverse_mod(a, n)¶

Returns the inverse of a mod n provided that a is prime with n.

EXAMPLES:

simula : inverse_mod(2, 5)
3
simula : inverse_mod(4, 7)
2

simula.ntheory.functions.is_prime(n)¶

Tests if n is a prime number.

Parameters: n – a nonnegative integer

EXAMPLES:

simula : is_prime(5)
True
simula : is_prime(201)
False

simula.ntheory.functions.is_primitive_root(a, p)¶

Tests if a is a primitive root of p.

Parameters

a – an integer
p – a prime number.

EXAMPLES:

simula : is_primitive_root(2, 5)
True
simula : is_primitive_root(3, 11)
False

simula.ntheory.functions.is_quad_residue(a, p)¶

Tests if a modulo p is in the set of squares mod p`.

Parameters

a – an integer
p – a prime number.

EXAMPLES:

simula : is_quad_residue(2, 5)
False
simula : is_quad_residue(3, 11)
True

simula.ntheory.functions.jacobi_symbol(m, n)¶

Returns the Jacobi symbol (m / n).

Parameters

m – an integer
n – an odd positive integer

EXAMPLES:

simula : jacobi_symbol(7, 15)
-1
simula : jacobi_symbol(2, 33)
1

simula.ntheory.functions.legendre_symbol(a, p)¶

Returns the Legendre symbol (m / n).

Parameters

m – an integer
n – an odd prime number

EXAMPLES:

simula : legendre_symbol(7, 11)
-1
simula : legendre_symbol(3, 37)
1

simula.ntheory.functions.list_divisors(n, *, proper=False)¶

Return all divisors of n sorted from 1 to n.

Parameters

n – an integer
proper – (a boolean, default False) specify if n is included to the list o divisors or not.

EXAMPLES:

simula : list_divisors(10)
[1, 2, 5, 10]
simula : list_divisors(10, proper=True)
[1, 2, 5]

simula.ntheory.functions.loggamma(n)¶

Returns log(gamma(n)).

EXAMPLES:

simula : loggamma(1)
0
simula : loggamma(2)
log(2)

simula.ntheory.functions.mobius(n)¶

Returns mobius(n) which maps natural number to {-1, 0, 1}.

Parameters: n – a positive integer

EXAMPLES:

simula : mobius(2)
-1
simula : mobius(15)
1

simula.ntheory.functions.multiplicity(m, n)¶

Returns the greatest integer k such that m^k divides n.

Parameters

m – an integer
n – an integer

EXAMPLES:

simula : multiplicity(10, 100)
2
simula : multiplicity(3, 36)
2

simula.ntheory.functions.n(x, *, precision=15)¶

Returns a numerical approximation of x for a given precision precision (default : 15).

EXAMPLES :

simula : numerical_approx(pi)
3.141592653589793
simula : numerical_approx(pi, precision=20)
3.14159265358979323846

simula.ntheory.functions.nAk(n, k)¶: Returns the k-arrangement in n i.e. \(\dfrac{n!}{(n-k)!}\).

simula.ntheory.functions.nCk(n, k)¶: Returns the binomial coefficient \(\dfrac{n!}{(n-k)! \times k! }\).

simula.ntheory.functions.next_prime(n)¶

Returns the i-th prime number greater than n.

Parameters: n – an integer

EXAMPLES:

simula : next_prime(4)
5
simula : next_prime(23)
29

simula.ntheory.functions.nthroot_mod(a, n, p, *, all_roots=False)¶

Returns the solutions to x^n = a mod p.

Parameters

a – an integer
n – a positive integer
p – a positive integer
all_roots – (default False) if False returns the smallest root, else the list of roots

EXAMPLES:

simula : nthroot_mod(1, 2, 7)
1
simula : nthroot_mod(1, 2, 7, all_roots=True)
[1, 6]

simula.ntheory.functions.number_divisors(n, *, proper=False)¶

Return the number of divisors of n.

Parameters

n – an integer
proper – (default False) If True then the divisor of n will not be counted

EXAMPLES:

simula : number_divisors(10)
4
simula : number_divisors(10, proper=True)
3

simula.ntheory.functions.numerator(expr)¶

Returns the numerator of expr.

EXAMPLES:

simula : f = 2/(sqrt(3)-1); f
2/(-1 + sqrt(3))
simula : numerator(f)
2
simula : numerator(3pi/5)
3pi

simula.ntheory.functions.numerical_approx(x, *, precision=15)¶

Returns a numerical approximation of x for a given precision precision (default : 15).

EXAMPLES :

simula : numerical_approx(pi)
3.141592653589793
simula : numerical_approx(pi, precision=20)
3.14159265358979323846

simula.ntheory.functions.order_modulo(a, n)¶

Returns the order of a modulo n.

Parameters

a – an integer
n – an integer relatively prime to a

EXAMPLES:

simula : order_modulo(2, 9)
6
simula : order_modulo(3, 11)
5

simula.ntheory.functions.perfect_power(n)¶

Returns (a, e) such that \(n=a^e\) if n is a perfect power with e > 1, else False.

Parameters: n – an integer

EXAMPLES:

simula : perfect_power(6)
False
simula : perfect_power(100)
(10, 2)
simula : perfect_power(16807)
(7, 5)

simula.ntheory.functions.power_mod(x, a, n)¶

Returns the power x^a mod n.

EXAMPLES:

simula : power_mod(100, 10000, 11)
1
simula : power_mod(99, 99876655, 13)
5

simula.ntheory.functions.previous_prime(n)¶

Returns the i-th prime number less than n.

Parameters: n – an integer

EXAMPLES:

simula : previous_prime(4)
3
simula : previous_prime(23)
19

simula.ntheory.functions.prime_factors(n)¶

Returns a sorted list of n’s prime factors, ignoring multiplicity.

Parameters: n – an integer

EXAMPLES:

simula : prime_factors(6)
[2, 3]
simula : prime_factors(20)
[2, 5]

simula.ntheory.functions.prime_pi(n)¶

Returns pi(n) the number of prime numbers less than or equal to n.

Parameters: n – an integer

EXAMPLES:

simula : prime_pi(4)
2
simula : prime_pi(20)
8

simula.ntheory.functions.prime_position(n)¶

Returns the n-th prime number.

Parameters: n – a positive integer

EXAMPLES:

simula : prime_position(1)
2
simula : prime_position(2)
3
simula : prime_position(10)
29

simula.ntheory.functions.prime_range(a, b)¶: Returns the list of primes between a and b.

simula.ntheory.functions.primes(start, end)¶

Generators of primes between start and end (both included).

EXAMPLES:

simula : 11 in primes(10, 40)
True
simula : for i in primes(10, 40):
. . . . . . :     print(i)
11
13
17
19
23
29
31
37

simula.ntheory.functions.primitive_root(n)¶

Returns the smallest primitive root modulo n or Raise an exception valueError.

EXAMPLES:

simula : primitive_root(7)
3
simula : primitive_root(10)
3
simula : primitive_root(29)
2

simula.ntheory.functions.primitive_root_mod(n)¶

Returns the smallest primitive root or None.

Parameters: n – a positive integer

EXAMPLES:

simula : primitive_root_mod(4)
3
simula : primitive_root_mod(20)
simula : primitive_root_mod(19)
2

simula.ntheory.functions.quadratic_residues(n)¶

Returns the set of quadratic residues mod n.

Parameters: n – a positive integer

EXAMPLES:

simula : quadratic_residues(4)
{0, 1}
simula : quadratic_residues(20)
{0, 1, 4, 5, 9, 16}

simula.ntheory.functions.randint(a, b=None)¶

Returns a random integer between a and b if b is not None` otherwise between ``0 and ``a`.

Parameters: b (Optional[int]) –

simula.ntheory.functions.random_prime(a, b=None)¶

Returns randomly a prime number between a and b.

Parameters: b (Optional[int]) –

simula.ntheory.functions.random_prime_size(size)¶

Returns randomly a prime of size size bits.

Parameters: size (int) –

simula.ntheory.functions.rationalize_denominator(expr)¶

Rationalizes the denominator of expr.

EXAMPLES:

simula : f = 2/(sqrt(3)-1); f
2/(-1 + sqrt(3))
simula : rationalize_denominator(f)
1 + sqrt(3)
simula : rationalize_denominator(1/(5-sqrt(11)))
(sqrt(11) + 5)/14

Parameters: expr (sympy.core.expr.Expr) –

simula.ntheory.functions.sign(x)¶

Returns the sign of x.

EXAMPLES:

simula :  sign(2-pi)
-1
simula : sign(2-sqrt(3))
1

simula.ntheory.functions.sqrt_mod(a, n, all_roots=False)¶

Returns a root of \(x^2 = a \mod n\) or None.

Parameters

a – an integer
n – a positive integer
all_roots – (default False) if True the list of roots is returned or None

EXAMPLES:

simula : sqrt_mod(4, 7)
2
simula : sqrt_mod(4, 7, all_roots=True)
[2, 5]
simula : sqrt_mod(5, 11, all_roots=True)
[4, 7]
simula : sqrt_mod(5, 10, all_roots=True)
[5]

simula.ntheory.functions.srange(start, stop=None, step=1)¶

Returns all integers from start to stop provided that stop is not None otherwise returns all integers from 0 to start for a given step` (default is 1).

EXAMPLES:

simula : srange(2, 9)
[2, 3, 4, 5, 6, 7, 8]
simula : srange(9, 0, -1)
[9, 8, 7, 6, 5, 4, 3, 2, 1]

Complex Numbers¶

Functions acting on Complex numbers

simula.ntheory.complexe.argument(z)¶

return the argument of z.

Parameters: z – a complex number

EXAMPLES:

simula : argument(1-I)
-pi/4
simula : argument(I)
pi/2

simula.ntheory.complexe.complex_alg_form(z)¶

return the algebraic form of z.

Parameters: z – a complex number

EXAMPLES:

simula : z = 1/(1 - I); z
(1 + I)/2
simula : complex_alg_form(z)
1/2 + I/2

simula.ntheory.complexe.complex_exp_form(z)¶

return the exponential form of z.

Parameters: z – a complex number

EXAMPLES:

simula : complex_exp_form(1 - I)
'sqrt(2)*exp(-I*pi/4)'
simula : complex_exp_form(-I)
'exp(-I*pi/2)'

simula.ntheory.complexe.complex_trig_form(z)¶

return the trigonometric form of z.

Parameters: z – a complex number

EXAMPLES:

simula : complex_trig_form(1 - I)
'sqrt(2)(cos(-pi/4)+I*sin(-pi/4))'
simula : complex_trig_form(-I)
'cos(-pi/2)+I*sin(-pi/2)'

simula.ntheory.complexe.complexe(a, b=0)¶

return the complex number a + bI.

Parameters

a (int) – an real number
b – an real number

EXAMPLES:

simula : complex(2, 6)
2 + 6I
simula : complex(1, 1)
1 + I

simula.ntheory.complexe.conjugate(z)¶

return the conjugate of z.

Parameters: z – a complex number

EXAMPLES:

simula : conjugate(1-I)
1 + I
simula : conjugate(1-I)(6I)
-6I

simula.ntheory.complexe.im_part(z)¶

return the imaginary part of z.

Parameters: z – a complex number

EXAMPLES:

simula : im_part(1-I)
-1
simula : im_part(6I)
6

simula.ntheory.complexe.imag_part(z)¶

return the imaginary part of z.

Parameters: z – a complex number

EXAMPLES:

simula : im_part(1-I)
-1
simula : im_part(6I)
6

simula.ntheory.complexe.module(z)¶

return the module of z i.e. |z|.

Parameters: z – a complex number

EXAMPLES:

simula : module(1-I)
sqrt(2)
simula : module(-I)
1

simula.ntheory.complexe.real_part(z)¶

return the real part of z.

Parameters: z – a complex number

EXAMPLES:

simula : real_part(1-I)
1
simula : real_part(6I)
0