sample.py

# -*- coding: utf-8 -*-

from __future__ import print_function

import src as ntlib

print()

# Create an instacne of Polynomial.
print('Create Polynomial.')
poly_0 = ntlib.Polynomial(('b', (2, -1), (0, 1)), ('c', (7, 1), (1, -1)))
poly_1 = ntlib.Polynomial(('a', (1, 3), (3, 4), (2, 2), (34, (1+3j))))
poly_2 = ntlib.Polynomial(('a', (1, 0), (4, -4), (2, 3), (0, 1)))
poly_3 = poly_1 / poly_2
print(poly_0)
print(poly_1[::])
print(poly_3)

print()

# Create an instacne of Congruence.
print('Create Congruence.')
_con = ntlib.Congruence(('z', (2, 1), (0, -46)), mod=105)
_ret = _con.solution

print('The solution of %s is\n' % str(_con))
print('\t', _ret)

print()

# Create an instacne of Quadratic.
print('Create Quadratic.')
_qua = ntlib.Quadratic(8068, vars=('p', 'q'))
_rst = _qua.solution

print('The solution of %s is\n' % str(_qua))
print('\t', _rst)

print()

# Create an instacne of Index.
print('Create Index.')
index = ntlib.Index(41)
print(index)
print()
rst = index(4, 8)
print('4 * 8 ≡ %d' % rst)

print()

# Create an instacne of Legendre.
print('Create Legendre.')
l1 = ntlib.Legendre(3, 17)
l2 = ntlib.Legendre('3|17')
l3 = ntlib.Legendre(l1)

print(l1, l1.eval())
print(l2, l2.simplify())
print(l3, l3.reciprocate())

print()

# Create an instacne of Jacobi.
print('Create Jacobi.')
j1 = ntlib.Jacobi(47, 359)
j2 = ntlib.Jacobi('47|359')
j3 = ntlib.Jacobi(l1)

print(j1, j1.eval())
print(j2, j2.simplify())
print(j3, j3.reciprocate())

print()

# Create an instacne of Fraction.
print('Create Fraction.')
print('7700/2145 = ', end=' ')
rst_ = ntlib.Fraction('7699/2145')
dst_ = ntlib.Fraction(1, 2145)
print(rst_ + dst_)

print()

# Print all primes numbers less than 97.
print('Call primelist.')
print(ntlib.primelist(97, -1))

print()

# Check if 13 and 20 are divisible.
print('Call isdivisible.')
if ntlib.isdivisible(13, 20):
    print('The result is 13|20.')
else:
    print('The result is 13∤20.')

print()

# Check if 197 is prime.
print('Call isprime.')
if ntlib.isprime(197):
    print('197 is a prime number.')
else:
    print('197 is a composit number.')

print()

# Print the GCD of 10 and 24.
print('Call gcd.')
print('(10,24) = %d' % ntlib.gcd(10, 24))

print()

# Check if 157 and 673 are coprime numbers.
print('Call coprime.')
if ntlib.coprime(157, 673):
    print('157 and 673 are coprime.')
else:
    print('157 and 673 are not coprime.')

print()

# Print the Bézout Equation with parameters 179 and 367.
print('Call bezout.')
print('%d*179 + %d*367 = (179,367)' % ntlib.bezout(179, 367))

print()

# Print the factorisation of -234.
print('Call factor.')
print('-100 =', end=' ')
ctr = 0
for prime in ntlib.factor(-100):
    if ctr > 0:
        print('*', end=' ')
    print(prime, end=' ')
    ctr += 1
print()

print()

# Print the solutions for N|a^2-b^2 while N∤a+b and N∤a-b.
print('Call decomposit.')
print('The solutions for N|a^2-b^2 while N∤a+b and N∤a-b\
        is\n\ta = %d\n\tb = %d' % ntlib.decomposit(100))

print()

# Return the special solutions for indefinite binary equation, 7*x + 24*y = -3.
print('Call binary.')
print('The general solutions for \'7*x + 24*y = -3\' is (t∈Z)\
        \n\tx = %d + 24*t\n\ty = %d - 7*t' % ntlib.binary(7, 24, -3))

print()

# Print the result of 235^12235 (mod 9).
print('Call modulo.')
print('235 ^ 12235 ≡ %d (mod 9)\n' % ntlib.modulo(235, 12235, 9))

print()

# Return the Euclidean division result of
# (20140515x^20140515 + 201495x^201405 + 2014x^2014
# + 8x^8 + x^6 + 4x^3 + x + 1) / (x^7 - 1).
print('Call polydiv.')
dvdExp = [1, 3, 2, 34]
dvdCoe = [-2, 4, 3, 1]
dvsExp = [7,  1]
dvsCoe = [1, -1]

(qttExp, qttCoe, rtoExp, rtoCoe) = ntlib.polydiv(
                        dvdExp, dvdCoe, dvsExp, dvsCoe)

print('\n\t')
for ptr in range(len(dvdExp)):
    print('%dx^%d' % (dvdCoe[ptr], dvdExp[ptr]), end=' ')
    if ptr < len(dvdExp) - 1:
        print('+', end=' ')
print('=\n(', end=' ')
for ptr in range(len(dvsExp)):
    print('%dx^%d' % (dvsCoe[ptr], dvsExp[ptr]), end=' ')
    if ptr < len(dvsExp) - 1:
        print('+', end=' ')
print(') * (', end=' ')
for ptr in range(len(qttExp)):
    print('%dx^%d' % (qttCoe[ptr], qttExp[ptr]), end=' ')
    if ptr < len(qttExp) - 1:
        print('+', end=' ')
print(') +', end=' ')
for ptr in range(len(rtoExp)):
    print('%dx^%d' % (rtoCoe[ptr], rtoExp[ptr]), end=' ')
    if ptr < len(rtoExp) - 1:
        print('+', end=' ')
print()

print()

# Return the coefficients and exponents of quotient and remainder
# after congruence simplification of
#   3x^14 + 4x^13 + 2x^11 + x^9 + x^6 + x^3 + 12x^2 + x ≡ 0 (mod 5).
print('Call simplify.')
cgcExp = [14, 13, 11,  9,  6,  3,  2,  1]
cgcCoe = [3,  4,  2,  1,  1,  1, 12,  1]
modulo = 5

(rtoExp, rtoCoe) = ntlib.simplify(cgcExp, cgcCoe, modulo)

print('The original polynomial congruence is\n\t', end=' ')
for ptr in range(len(cgcExp)):
    print('%dx^%d' % (cgcCoe[ptr], cgcExp[ptr]), end=' ')
    if ptr < len(cgcExp) - 1:
        print('+', end=' ')
print('≡ 0 (mod %d)' % modulo)
print()
print('The simplified polynomial congruence is\n\t', end=' ')
for ptr in range(len(rtoExp)):
    print('%dx^%d' % (rtoCoe[ptr], rtoExp[ptr]), end=' ')
    if ptr < len(rtoExp) - 1:
        print('+', end=' ')
print('≡ 0 (mod %d)' % modulo)

print()

# Return the solutions of a naïve congruence set.
print('Call crt')
remainder = ntlib.crt((3, [1, -1]), (5, [1, -1]), (7, [2, -2]))

print('x ≡ ±1 (mod 3)')
print('x ≡ ±1 (mod 5)')
print('x ≡ ±2 (mod 7)')
print('The solutions of the above equation set is\n\tx ≡', end=' ')
for rst in remainder:
    print(rst, end=' ')
print('(mod 105)')

print()

# Return the solutions of congruence equation x^2 ≡ 46 (mod 105).
print('Call congsolve')
cgcExp = [2, 0]
cgcCoe = [1, -46]
modulo = 105

for ptr in range(len(cgcExp)):
    print('%dx^%d' % (cgcCoe[ptr], cgcExp[ptr]), end=' ')
    if ptr < len(cgcExp) - 1:
        print('+', end=' ')
print('≡ 0 (mod %d)' % modulo)

remainder = ntlib.congsolve(cgcExp, cgcCoe, modulo)

print('The solution of the above polynomial congruence is\n\tx ≡', end=' ')
for rst in remainder:
    print('%d' % rst, end=' ')
print('(mod %d)' % modulo)

print()

# Return the solutions of quadratic equation x^2 + y^2 = 2017.
print('Call quadratic.')
print('The solution of the equation x^2 + y^2 = 2017 is\
        \n\tx=±%d, y=±%d' % ntlib.quadratic(2017))

print()

# Return the order of 2 for modulo 9.
print('Call ord.')
print('The order of 2 mod 9 is\n\tord_9(2) = %d' % ntlib.ord(9, 2))

print()

# Return Euler function φ(40)
print('Call euler.')
print('φ(40) = %d' % ntlib.euler(40))

print()

# Return the primitive residue class of integer 40.
print('Call prc')
print('The primitive residue class of 40 is')
for num in ntlib.prc(40):
    print(num, end=' ')
print('.')

print()

# Return the primitive root(s) of modulo 7.
print('Call root.')
print('The primtive root(s) of modulo 7 is/are', end=' ')
for root in ntlib.root(7):
    print(root, end=' ')
print('.')

print()

# Return the result of Legendre symbol for (3 | 17).
print('Call legendre.')
print('(3 | 17) = %d' % ntlib.legendre(3, 17))

print()

# Return the result of jocabi symbol for (47 | 359).
print('Call jacobi.')
print('(47 | 359) = %d' % ntlib.jacobi(47, 359))

print()

# Return if an integer is a Carmicheal number.
print('Call carmicheal.')
print('3499', end=' ')
if ntlib.carmicheal(3499):
    print('is', end=' ')
else:
    print('isn\'t', end=' ')
print('a Carmicheal number.')

print()

# Return a pseudo-prime number with certain paterns.
print('Call pseudo.')
print('A pseudo-prime number with Fermat test (t=100000) is %d.'
    % ntlib.pseudo(byte=16, para=10000, mode='Fermat', flag=False))
print('An Euler pseudo-prime number with Solovay-Stassen test (t=100000) is %d.'
    % ntlib.pseudo(byte=16, para=10000, mode='Euler'))
print('A strong pseudo-prime number with Miller-Rabin test (k=100000) is %d.'
    % ntlib.pseudo(byte=16, para=10000, mode='Strong'))

print()

# Return the continued fraction of a number.
print('Call fraction.')
fct_ = __import__('fractions').Fraction(7700, 2145)
rst_ = ntlib.fraction(fct_)
print(fct_, '=', rst_)

print()