def totient_comparisons(max_n):
#x is our modulus
false_pos = 0
false_neg = 0
for x in range(3, max_n):
self_squares = []
#factoring to find invertible numbers
primes = factorint(x)
#checking all the numbers where gcd(x,y)=1
for y in range(1, x):
#checks that no prime divisors of our modulus (x) divide our number y
#then checks if y is its own inverse
if all(y % p != 0 for p in primes.keys()) and (y * y) % x == 1:
self_squares.append(y)
#if we have some interesting solutions, verify the relationship between
#totient(n)%4 and the existence of solutions (it's necessary, not sufficient)
if (len(self_squares) > 2) != (sp.totient(x) % 4 == 0):
if len(self_squares) > 2:
false_neg += 1
else:
false_pos += 1
if len(self_squares) > 2:
print("Solution totient: {}".format(
sp.factorint(sp.totient(max(self_squares)))))
#looking at the factorizations of solutions
print("N totient: {}".format(sp.factorint(sp.totient(x))))
print("False positive solns: {} False negative solns: {}".format(
false_pos, false_neg))
def get_reps(n, coprime=True):
# note that each representative represents n-1 other entries
# so in total there are phi(2**n - 1)
l = sympy.totient(2**n - 1) / n
reps = [1]
p = 2
while p < 2**n - 1:
if coprime:
m = p = sympy.nextprime(p)
else:
p += 2
if p % 2 == 0:
p -= 1
if coprime and gcd(p, 2**n - 1) != 1:
continue
i = 1
for i in range(1, n):
if (p * 2**i) % (2**n - 1) in reps:
break
if coprime:
m = m if (m < (p * 2**i) % (2**n - 1)) else ((p * 2**i) %
(2**n - 1))
if (p * 2**i) % (2**n - 1) in reps:
continue
if not coprime:
m = p
reps.append(m)
if len(reps) == l and coprime:
break
return sorted(reps)
def order(n, elt):
if elt in coPrime(n):
for i in sorted(divisors(totient(n))):
if elt ** i % n == 1:
return i
else:
return "Element not in set"
def magic_gcd(m, n):
ans = 0
for k in xrange(1, n+1):
if m % k == 0 and n % k == 0:
print k
ans += totient(k)
return ans
def totient_chain_length(n):
global cache
if n == 1:
return 1
if n not in cache:
cache[n] = totient_chain_length(totient(n))
return 1 + cache[n]
def euler69():
ret = 0
max_calc = 0
for n in xrange(2, 1000001):
t = n * 1.0 / totient(n)
if max(max_calc, t) == t:
max_calc = t
ret = n
return ret
def R(d):
"""
The number of resilient fractions can be given as the Euler totient of the
given number n as we are looking for the number of coprime integers less
than n.
So R(d) = totient(d) / (d-1)
"""
return Fraction(totient(d), d - 1)
def min_totient_ratio(arr):
ret, min_ratio = 0, 1000
for n in arr:
t = totient(n)
if is_permutation(n, t):
ratio = n * 1.0 / t
if ratio < min_ratio:
min_ratio = ratio
ret = n
return min_ratio, ret
def grapher_tau(n):
import matplotlib.pyplot as plt
X=[]
Y=[]
for i in range(1,n+1):
X.append(i)
Y.append(tau(i)+sp.totient(i))
if(i%500==0):
print(i," segment "," covered")
plt.plot(X,Y)
def G(N):
s1, s2 = 0, 0
for d in range(1, math.floor(math.sqrt(N)) + 1):
s1 += (d * H(math.floor(N / d))) % 998244353
#print(d, s1)
print("1st Sum Completed...")
for c in range(1, math.floor(math.sqrt(N)) + 1):
s2 += (sympy.totient(c) * F(math.floor(N / c))) % 998244353
#print(c, s2)
print("2nd Sum Completed...")
s3 = (F(math.floor(math.sqrt(N))) *
H(math.floor(math.sqrt(N)))) % 998244353
return (s1 + s2 - s3) % 998244353
def calc_pn(n):
n1 = len(n)
phi_n = sympy.totient(n1)
for i in range(1, phi_n + 1):
myk = (2**i) % n1
if myk == 1:
ord_2 = i
a = ord_2 / 2
myk2 = (2**a) % n1
if myk2 == n1 - 1:
pn = n1 * (2**a - 1)
else:
pn = 2**a - 1
return pn
from sympy import totient, mod_inverse
# n = 0, 1, 2, 3
result = 1 + 3 + 7 + 61
p = 2**8
q = 7**8
modulo = p * q
phi_chain = [q]
k = q
while k > 1:
k = totient(k)
phi_chain.append(k)
# A(n, n) + 3 = 0 (mod 2^8) for n > 3
u = p * mod_inverse(p, q)
# n = 4 A(4, 4) = 2^2^2^65536 - 3
w = 65536
for i in 2, 1, 0:
w = pow(2, w, phi_chain[i])
result = (result + w * u - 3) % modulo
# n = 5, 6 A(5, 5) % modulo = A(6, 6) % modulo
w = 2
for phi in reversed(phi_chain):
w = pow(2, w, phi)
result = (result + 2 * (w * u - 3)) % modulo
print(result)
for p in PRIMES:
if p > halfN:
break
div, mod = divmod(n, p)
if mod == 0:
factors.append(p)
prevDiv = div
div, mod = divmod(div, p)
while mod == 0:
prevDiv = div
div, mod = divmod(div, p)
factors += factorsOf(prevDiv)
return set(factors)
def nOnPhiN(n):
# p = factorint(n).keys()
p = factorsOf(n)
total = 1
for fact in p:
total *= (fact - 1) / fact
return 1 / total
possible = sorted([(nOnPhiN(n), n) for n in trange(2, top)], key=lambda a: (a[0], top - a[1]))
for pair in tqdm(possible):
ratio, n = pair
if is_perm(n, totient(n)):
print(n)
break
p = 12039102490128509125925019010000012423515617235219127649182470182570195018265927223
q = 1039300813886545966418005631983853921163721828798787466771912919828750891
assert (p - 1) % q == 0
g = 10729072579307052184848302322451332192456229619044181105063011741516558110216720725
x = int.from_bytes(b"Hi! I am Vadim Davydov from ITMO University", 'big')
G = pow(g, x, p)
Gp = int.to_bytes(G, ceil(log2(G) / 8), 'big')
a = sha512(Gp).digest()
ap = int.from_bytes(a, 'big')
t = int(totient(p))
aa = pow(ap, ap, t)
h = aa % p
print("p =", p)
print("q =", q)
print("g =", g)
print("x =", x)
print("G =", G)
print("G'=", Gp)
print("a =", a)
print("a'=", ap)
print("t =", t)
print("aa=", aa)
# Number of test cases
size = 250
# Cardinality of the set {m : phi(m) <= N}
C_N = np.zeros(size - 1)
# Values of N to be tested
Ns = [i for i in range(1, size)]
# Set of all values m s.t. phi(m) <= N, we do not need to recalcuate these values
# if phi(m) <= N then, phi(m) <= N + 1 and so on
ms = {1}
for N in range(1, size):
# Checks all values up to our analytic upper bound
for m in range(1, 2 * N**2):
# Skips values of m that have already been calculated
if m in ms:
continue
elif sp.totient(m) <= N:
ms.add(m)
C_N[N - 1] += len(ms)
# Stores pairs N and C_N when C_N > 2N (our trent line)
pairs = []
for i in range(len(Ns)):
if C_N[i] > 2 * Ns[i]:
pairs.append((Ns[i], C_N[i]))
# Plots data and lines
plt.figure(figsize=(30, 15))
plt.plot([0, size], [0, size], 'k-')
plt.plot([0, size], [0, 2 * size], 'g-')
plt.plot([0, size], [0, 3 * size], 'k-')
plt.bar(Ns, C_N)
plt.show()
from numpy import argmax from sympy import totient from tqdm import trange print(argmax([n / totient(n) for n in trange(1, 1_000_000)]) + 1)
#!/usr/bin/python3
# -*- coding: utf-8 -*-
# https://projecteuler.net/problem=70
from sympy import totient
LIMIT = 10**7
oran = 42
for n in range(2,LIMIT):
to = totient(n)
for p in set(str(n)):
if(str(n).count(p)!=str(to).count(p)):
break
else:
if(set(str(n))==set(str(to))):
if(len(set(str(n)))==len(set(str(to)))):
if(n/to<oran):
oran = n/to
enkucuk = n
print(enkucuk)
def euler72():
ret = 0
for d in xrange(2, 1000001):
ret += totient(d)
return ret
# -*- coding: utf-8 -*-
"""
Created on Sat Mar 05 19:13:49 2016
@author: Lucas
"""
import fractions
import lucaslib
import sympy
def phi(n):
result = 0
for i in xrange(1,n+1):
if fractions.gcd(n,i) == 1:
result += 1
return result
maxval = 2
for x in xrange(2,1000000):
if float(x)/sympy.totient(x) > maxval:
maxval = float(x)/sympy.totient(x)
print x
print float(x)/sympy.totient(x)
print
def number_reduced_proper_fractions(max_denominator):
return sum(totient(d) for d in tqdm(range(2, max_denominator + 1)))
def test_latex_functions():
assert latex(exp(x)) == "e^{x}"
assert latex(exp(1) + exp(2)) == "e + e^{2}"
f = Function('f')
assert latex(f(x)) == '\\operatorname{f}{\\left (x \\right )}'
beta = Function('beta')
assert latex(beta(x)) == r"\beta{\left (x \right )}"
assert latex(sin(x)) == r"\sin{\left (x \right )}"
assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
assert latex(sin(2*x**2), fold_func_brackets=True) == \
r"\sin {2 x^{2}}"
assert latex(sin(x**2), fold_func_brackets=True) == \
r"\sin {x^{2}}"
assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}"
assert latex(asin(x)**2, inv_trig_style="full") == \
r"\arcsin^{2}{\left (x \right )}"
assert latex(asin(x)**2, inv_trig_style="power") == \
r"\sin^{-1}{\left (x \right )}^{2}"
assert latex(asin(x**2), inv_trig_style="power",
fold_func_brackets=True) == \
r"\sin^{-1} {x^{2}}"
assert latex(factorial(k)) == r"k!"
assert latex(factorial(-k)) == r"\left(- k\right)!"
assert latex(subfactorial(k)) == r"!k"
assert latex(subfactorial(-k)) == r"!\left(- k\right)"
assert latex(factorial2(k)) == r"k!!"
assert latex(factorial2(-k)) == r"\left(- k\right)!!"
assert latex(binomial(2, k)) == r"{\binom{2}{k}}"
assert latex(
FallingFactorial(3, k)) == r"{\left(3\right)}_{\left(k\right)}"
assert latex(RisingFactorial(3, k)) == r"{\left(3\right)}^{\left(k\right)}"
assert latex(floor(x)) == r"\lfloor{x}\rfloor"
assert latex(ceiling(x)) == r"\lceil{x}\rceil"
assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)"
assert latex(Min(x, y)**2) == r"\min\left(x, y\right)^{2}"
assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)"
assert latex(Max(x, y)**2) == r"\max\left(x, y\right)^{2}"
assert latex(Abs(x)) == r"\left\lvert{x}\right\rvert"
assert latex(re(x)) == r"\Re{x}"
assert latex(re(x + y)) == r"\Re{x} + \Re{y}"
assert latex(im(x)) == r"\Im{x}"
assert latex(conjugate(x)) == r"\overline{x}"
assert latex(gamma(x)) == r"\Gamma\left(x\right)"
assert latex(Order(x)) == r"\mathcal{O}\left(x\right)"
assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'
assert latex(cot(x)) == r'\cot{\left (x \right )}'
assert latex(coth(x)) == r'\coth{\left (x \right )}'
assert latex(re(x)) == r'\Re{x}'
assert latex(im(x)) == r'\Im{x}'
assert latex(root(x, y)) == r'x^{\frac{1}{y}}'
assert latex(arg(x)) == r'\arg{\left (x \right )}'
assert latex(zeta(x)) == r'\zeta\left(x\right)'
assert latex(zeta(x)) == r"\zeta\left(x\right)"
assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)"
assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)"
assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)"
assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)"
assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)"
assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)"
assert latex(
polylog(x, y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)"
assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)"
assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)"
assert latex(elliptic_k(z)) == r"K\left(z\right)"
assert latex(elliptic_k(z)**2) == r"K^{2}\left(z\right)"
assert latex(elliptic_f(x, y)) == r"F\left(x\middle| y\right)"
assert latex(elliptic_f(x, y)**2) == r"F^{2}\left(x\middle| y\right)"
assert latex(elliptic_e(x, y)) == r"E\left(x\middle| y\right)"
assert latex(elliptic_e(x, y)**2) == r"E^{2}\left(x\middle| y\right)"
assert latex(elliptic_e(z)) == r"E\left(z\right)"
assert latex(elliptic_e(z)**2) == r"E^{2}\left(z\right)"
assert latex(elliptic_pi(x, y, z)) == r"\Pi\left(x; y\middle| z\right)"
assert latex(elliptic_pi(x, y, z)**2) == \
r"\Pi^{2}\left(x; y\middle| z\right)"
assert latex(elliptic_pi(x, y)) == r"\Pi\left(x\middle| y\right)"
assert latex(elliptic_pi(x, y)**2) == r"\Pi^{2}\left(x\middle| y\right)"
assert latex(Ei(x)) == r'\operatorname{Ei}{\left (x \right )}'
assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left (x \right )}'
assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)'
assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left (x \right )}'
assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left (x \right )}'
assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left (x \right )}'
assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}{\left (x \right )}'
assert latex(
jacobi(n, a, b, x)) == r'P_{n}^{\left(a,b\right)}\left(x\right)'
assert latex(jacobi(n, a, b, x)**2) == r'\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}'
assert latex(
gegenbauer(n, a, x)) == r'C_{n}^{\left(a\right)}\left(x\right)'
assert latex(gegenbauer(n, a, x)**2) == r'\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(chebyshevt(n, x)) == r'T_{n}\left(x\right)'
assert latex(
chebyshevt(n, x)**2) == r'\left(T_{n}\left(x\right)\right)^{2}'
assert latex(chebyshevu(n, x)) == r'U_{n}\left(x\right)'
assert latex(
chebyshevu(n, x)**2) == r'\left(U_{n}\left(x\right)\right)^{2}'
assert latex(legendre(n, x)) == r'P_{n}\left(x\right)'
assert latex(legendre(n, x)**2) == r'\left(P_{n}\left(x\right)\right)^{2}'
assert latex(
assoc_legendre(n, a, x)) == r'P_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_legendre(n, a, x)**2) == r'\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(laguerre(n, x)) == r'L_{n}\left(x\right)'
assert latex(laguerre(n, x)**2) == r'\left(L_{n}\left(x\right)\right)^{2}'
assert latex(
assoc_laguerre(n, a, x)) == r'L_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_laguerre(n, a, x)**2) == r'\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(hermite(n, x)) == r'H_{n}\left(x\right)'
assert latex(hermite(n, x)**2) == r'\left(H_{n}\left(x\right)\right)^{2}'
# Test latex printing of function names with "_"
assert latex(
polar_lift(0)) == r"\operatorname{polar\_lift}{\left (0 \right )}"
assert latex(polar_lift(
0)**3) == r"\operatorname{polar\_lift}^{3}{\left (0 \right )}"
assert latex(totient(n)) == r'\phi\left( n \right)'
#!/usr/bin/env python from sympy import primefactors, mod_inverse, totient c = 10247955272908064997771727568918647737311526262165262458875076213296879253353684001266750329 e = 65537 n = 14393188157100504374851319373504728765762087532393762203410288659620743314524046223483350199 phi = totient(n) d = mod_inverse(e, phi) flag = pow(c, d, n) print(flag.to_bytes(flag.bit_length() // 8 + 1, 'big').decode())
def phi(n):
tot = sympy.totient(n)
print("phi(", n, ") =", tot)
return tot
def NumOfDecimals(n):
if n == 1 or n % 2 == 0 or n % 5 == 0: return (0, 0)
phid = divisors(totient(n))
for m in phid:
if (10**m - 1) % n == 0: return (n, m)
def test_latex_functions():
assert latex(exp(x)) == "e^{x}"
assert latex(exp(1) + exp(2)) == "e + e^{2}"
f = Function('f')
assert latex(f(x)) == r'f{\left (x \right )}'
assert latex(f) == r'f'
g = Function('g')
assert latex(g(x, y)) == r'g{\left (x,y \right )}'
assert latex(g) == r'g'
h = Function('h')
assert latex(h(x, y, z)) == r'h{\left (x,y,z \right )}'
assert latex(h) == r'h'
Li = Function('Li')
assert latex(Li) == r'\operatorname{Li}'
assert latex(Li(x)) == r'\operatorname{Li}{\left (x \right )}'
beta = Function('beta')
# not to be confused with the beta function
assert latex(beta(x)) == r"\beta{\left (x \right )}"
assert latex(beta) == r"\beta"
assert latex(sin(x)) == r"\sin{\left (x \right )}"
assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
assert latex(sin(2*x**2), fold_func_brackets=True) == \
r"\sin {2 x^{2}}"
assert latex(sin(x**2), fold_func_brackets=True) == \
r"\sin {x^{2}}"
assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}"
assert latex(asin(x)**2, inv_trig_style="full") == \
r"\arcsin^{2}{\left (x \right )}"
assert latex(asin(x)**2, inv_trig_style="power") == \
r"\sin^{-1}{\left (x \right )}^{2}"
assert latex(asin(x**2), inv_trig_style="power",
fold_func_brackets=True) == \
r"\sin^{-1} {x^{2}}"
assert latex(factorial(k)) == r"k!"
assert latex(factorial(-k)) == r"\left(- k\right)!"
assert latex(subfactorial(k)) == r"!k"
assert latex(subfactorial(-k)) == r"!\left(- k\right)"
assert latex(factorial2(k)) == r"k!!"
assert latex(factorial2(-k)) == r"\left(- k\right)!!"
assert latex(binomial(2, k)) == r"{\binom{2}{k}}"
assert latex(FallingFactorial(3,
k)) == r"{\left(3\right)}_{\left(k\right)}"
assert latex(RisingFactorial(3, k)) == r"{\left(3\right)}^{\left(k\right)}"
assert latex(floor(x)) == r"\lfloor{x}\rfloor"
assert latex(ceiling(x)) == r"\lceil{x}\rceil"
assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)"
assert latex(Min(x, y)**2) == r"\min\left(x, y\right)^{2}"
assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)"
assert latex(Max(x, y)**2) == r"\max\left(x, y\right)^{2}"
assert latex(Abs(x)) == r"\left\lvert{x}\right\rvert"
assert latex(re(x)) == r"\Re{x}"
assert latex(re(x + y)) == r"\Re{x} + \Re{y}"
assert latex(im(x)) == r"\Im{x}"
assert latex(conjugate(x)) == r"\overline{x}"
assert latex(gamma(x)) == r"\Gamma{\left(x \right)}"
w = Wild('w')
assert latex(gamma(w)) == r"\Gamma{\left(w \right)}"
assert latex(Order(x)) == r"\mathcal{O}\left(x\right)"
assert latex(Order(x, x)) == r"\mathcal{O}\left(x\right)"
assert latex(Order(x, x, 0)) == r"\mathcal{O}\left(x\right)"
assert latex(Order(x, x,
oo)) == r"\mathcal{O}\left(x; x\rightarrow\infty\right)"
assert latex(
Order(x, x, y)
) == r"\mathcal{O}\left(x; \begin{pmatrix}x, & y\end{pmatrix}\rightarrow0\right)"
assert latex(
Order(x, x, y, 0)
) == r"\mathcal{O}\left(x; \begin{pmatrix}x, & y\end{pmatrix}\rightarrow0\right)"
assert latex(
Order(x, x, y, oo)
) == r"\mathcal{O}\left(x; \begin{pmatrix}x, & y\end{pmatrix}\rightarrow\infty\right)"
assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'
assert latex(cot(x)) == r'\cot{\left (x \right )}'
assert latex(coth(x)) == r'\coth{\left (x \right )}'
assert latex(re(x)) == r'\Re{x}'
assert latex(im(x)) == r'\Im{x}'
assert latex(root(x, y)) == r'x^{\frac{1}{y}}'
assert latex(arg(x)) == r'\arg{\left (x \right )}'
assert latex(zeta(x)) == r'\zeta\left(x\right)'
assert latex(zeta(x)) == r"\zeta\left(x\right)"
assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)"
assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)"
assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)"
assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)"
assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)"
assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)"
assert latex(polylog(x,
y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)"
assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)"
assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)"
assert latex(elliptic_k(z)) == r"K\left(z\right)"
assert latex(elliptic_k(z)**2) == r"K^{2}\left(z\right)"
assert latex(elliptic_f(x, y)) == r"F\left(x\middle| y\right)"
assert latex(elliptic_f(x, y)**2) == r"F^{2}\left(x\middle| y\right)"
assert latex(elliptic_e(x, y)) == r"E\left(x\middle| y\right)"
assert latex(elliptic_e(x, y)**2) == r"E^{2}\left(x\middle| y\right)"
assert latex(elliptic_e(z)) == r"E\left(z\right)"
assert latex(elliptic_e(z)**2) == r"E^{2}\left(z\right)"
assert latex(elliptic_pi(x, y, z)) == r"\Pi\left(x; y\middle| z\right)"
assert latex(elliptic_pi(x, y, z)**2) == \
r"\Pi^{2}\left(x; y\middle| z\right)"
assert latex(elliptic_pi(x, y)) == r"\Pi\left(x\middle| y\right)"
assert latex(elliptic_pi(x, y)**2) == r"\Pi^{2}\left(x\middle| y\right)"
assert latex(Ei(x)) == r'\operatorname{Ei}{\left (x \right )}'
assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left (x \right )}'
assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)'
assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left (x \right )}'
assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left (x \right )}'
assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left (x \right )}'
assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}{\left (x \right )}'
assert latex(Chi(x)) == r'\operatorname{Chi}{\left (x \right )}'
assert latex(jacobi(n, a, b,
x)) == r'P_{n}^{\left(a,b\right)}\left(x\right)'
assert latex(jacobi(
n, a, b,
x)**2) == r'\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}'
assert latex(gegenbauer(n, a,
x)) == r'C_{n}^{\left(a\right)}\left(x\right)'
assert latex(gegenbauer(
n, a,
x)**2) == r'\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(chebyshevt(n, x)) == r'T_{n}\left(x\right)'
assert latex(chebyshevt(n,
x)**2) == r'\left(T_{n}\left(x\right)\right)^{2}'
assert latex(chebyshevu(n, x)) == r'U_{n}\left(x\right)'
assert latex(chebyshevu(n,
x)**2) == r'\left(U_{n}\left(x\right)\right)^{2}'
assert latex(legendre(n, x)) == r'P_{n}\left(x\right)'
assert latex(legendre(n, x)**2) == r'\left(P_{n}\left(x\right)\right)^{2}'
assert latex(assoc_legendre(n, a,
x)) == r'P_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_legendre(
n, a,
x)**2) == r'\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(laguerre(n, x)) == r'L_{n}\left(x\right)'
assert latex(laguerre(n, x)**2) == r'\left(L_{n}\left(x\right)\right)^{2}'
assert latex(assoc_laguerre(n, a,
x)) == r'L_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_laguerre(
n, a,
x)**2) == r'\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(hermite(n, x)) == r'H_{n}\left(x\right)'
assert latex(hermite(n, x)**2) == r'\left(H_{n}\left(x\right)\right)^{2}'
theta = Symbol("theta", real=True)
phi = Symbol("phi", real=True)
assert latex(Ynm(n, m, theta, phi)) == r'Y_{n}^{m}\left(\theta,\phi\right)'
assert latex(
Ynm(n, m, theta,
phi)**3) == r'\left(Y_{n}^{m}\left(\theta,\phi\right)\right)^{3}'
assert latex(Znm(n, m, theta, phi)) == r'Z_{n}^{m}\left(\theta,\phi\right)'
assert latex(
Znm(n, m, theta,
phi)**3) == r'\left(Z_{n}^{m}\left(\theta,\phi\right)\right)^{3}'
# Test latex printing of function names with "_"
assert latex(
polar_lift(0)) == r"\operatorname{polar\_lift}{\left (0 \right )}"
assert latex(polar_lift(0)**
3) == r"\operatorname{polar\_lift}^{3}{\left (0 \right )}"
assert latex(totient(n)) == r'\phi\left( n \right)'
# some unknown function name should get rendered with \operatorname
fjlkd = Function('fjlkd')
assert latex(fjlkd(x)) == r'\operatorname{fjlkd}{\left (x \right )}'
# even when it is referred to without an argument
assert latex(fjlkd) == r'\operatorname{fjlkd}'
def R(n):
return totient(n)/(n-1)
# -*- coding: utf-8 -*-
"""
Created on Thu Sep 15 21:01:57 2016
@author: Lucas
"""
minsofar = 1.0008
import lucaslib
import sympy
for i in xrange(5000001, 10**7, 2):
if sorted(str(sympy.totient(i))) == sorted(str(i)):
if i / float(sympy.totient(i)) < minsofar:
print i
if lucaslib.isprime(i):
print "Prime!"
print i / float(sympy.totient(i))
print
minsofar = i / float(sympy.totient(i))
def max_totient_ratio(N):
return max((n for n in range(2, N + 1)), key=lambda n: n / totient(n))
def cum_phi_count(x):
total = 0
for i in trange(2, x + 1):
total += totient(i)
return total
from sympy import totient
maxn = 6
maxq = 3
for i in range(2, 1000000):
q = i / totient(i)
if q > maxq:
maxq = q
maxn = i
print("{} / {} = {}".format(maxn, totient(maxn), maxq))
def min_permutated_totient_ratio(N):
return min((n for n in tqdm(range(2, N + 1))
if sorted(str(n)) == sorted(str(totient(n)))),
key=lambda n: n / totient(n))
def test_F9():
assert totient(1776) == 576
def count_totients_div_4(max_n):
count = 0
for x in range(2, max_n):
if sp.totient(x) % 4 == 0:
count += 1
print("Totients divis by 4 up to {}: {}".format(max_n, count))
def test_latex_functions():
assert latex(exp(x)) == "e^{x}"
assert latex(exp(1) + exp(2)) == "e + e^{2}"
f = Function('f')
assert latex(f(x)) == r'f{\left (x \right )}'
assert latex(f) == r'f'
g = Function('g')
assert latex(g(x, y)) == r'g{\left (x,y \right )}'
assert latex(g) == r'g'
h = Function('h')
assert latex(h(x, y, z)) == r'h{\left (x,y,z \right )}'
assert latex(h) == r'h'
Li = Function('Li')
assert latex(Li) == r'\operatorname{Li}'
assert latex(Li(x)) == r'\operatorname{Li}{\left (x \right )}'
beta = Function('beta')
# not to be confused with the beta function
assert latex(beta(x)) == r"\beta{\left (x \right )}"
assert latex(beta) == r"\beta"
assert latex(sin(x)) == r"\sin{\left (x \right )}"
assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
assert latex(sin(2*x**2), fold_func_brackets=True) == \
r"\sin {2 x^{2}}"
assert latex(sin(x**2), fold_func_brackets=True) == \
r"\sin {x^{2}}"
assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left (x \right )}"
assert latex(asin(x)**2, inv_trig_style="full") == \
r"\arcsin^{2}{\left (x \right )}"
assert latex(asin(x)**2, inv_trig_style="power") == \
r"\sin^{-1}{\left (x \right )}^{2}"
assert latex(asin(x**2), inv_trig_style="power",
fold_func_brackets=True) == \
r"\sin^{-1} {x^{2}}"
assert latex(factorial(k)) == r"k!"
assert latex(factorial(-k)) == r"\left(- k\right)!"
assert latex(subfactorial(k)) == r"!k"
assert latex(subfactorial(-k)) == r"!\left(- k\right)"
assert latex(factorial2(k)) == r"k!!"
assert latex(factorial2(-k)) == r"\left(- k\right)!!"
assert latex(binomial(2, k)) == r"{\binom{2}{k}}"
assert latex(
FallingFactorial(3, k)) == r"{\left(3\right)}_{\left(k\right)}"
assert latex(RisingFactorial(3, k)) == r"{\left(3\right)}^{\left(k\right)}"
assert latex(floor(x)) == r"\lfloor{x}\rfloor"
assert latex(ceiling(x)) == r"\lceil{x}\rceil"
assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)"
assert latex(Min(x, y)**2) == r"\min\left(x, y\right)^{2}"
assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)"
assert latex(Max(x, y)**2) == r"\max\left(x, y\right)^{2}"
assert latex(Abs(x)) == r"\left\lvert{x}\right\rvert"
assert latex(re(x)) == r"\Re{x}"
assert latex(re(x + y)) == r"\Re{x} + \Re{y}"
assert latex(im(x)) == r"\Im{x}"
assert latex(conjugate(x)) == r"\overline{x}"
assert latex(gamma(x)) == r"\Gamma\left(x\right)"
assert latex(Order(x)) == r"\mathcal{O}\left(x\right)"
assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'
assert latex(cot(x)) == r'\cot{\left (x \right )}'
assert latex(coth(x)) == r'\coth{\left (x \right )}'
assert latex(re(x)) == r'\Re{x}'
assert latex(im(x)) == r'\Im{x}'
assert latex(root(x, y)) == r'x^{\frac{1}{y}}'
assert latex(arg(x)) == r'\arg{\left (x \right )}'
assert latex(zeta(x)) == r'\zeta\left(x\right)'
assert latex(zeta(x)) == r"\zeta\left(x\right)"
assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)"
assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)"
assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)"
assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)"
assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)"
assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)"
assert latex(
polylog(x, y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)"
assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)"
assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)"
assert latex(elliptic_k(z)) == r"K\left(z\right)"
assert latex(elliptic_k(z)**2) == r"K^{2}\left(z\right)"
assert latex(elliptic_f(x, y)) == r"F\left(x\middle| y\right)"
assert latex(elliptic_f(x, y)**2) == r"F^{2}\left(x\middle| y\right)"
assert latex(elliptic_e(x, y)) == r"E\left(x\middle| y\right)"
assert latex(elliptic_e(x, y)**2) == r"E^{2}\left(x\middle| y\right)"
assert latex(elliptic_e(z)) == r"E\left(z\right)"
assert latex(elliptic_e(z)**2) == r"E^{2}\left(z\right)"
assert latex(elliptic_pi(x, y, z)) == r"\Pi\left(x; y\middle| z\right)"
assert latex(elliptic_pi(x, y, z)**2) == \
r"\Pi^{2}\left(x; y\middle| z\right)"
assert latex(elliptic_pi(x, y)) == r"\Pi\left(x\middle| y\right)"
assert latex(elliptic_pi(x, y)**2) == r"\Pi^{2}\left(x\middle| y\right)"
assert latex(Ei(x)) == r'\operatorname{Ei}{\left (x \right )}'
assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left (x \right )}'
assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)'
assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left (x \right )}'
assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left (x \right )}'
assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left (x \right )}'
assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}{\left (x \right )}', latex(Chi(x)**2)
assert latex(
jacobi(n, a, b, x)) == r'P_{n}^{\left(a,b\right)}\left(x\right)'
assert latex(jacobi(n, a, b, x)**2) == r'\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}'
assert latex(
gegenbauer(n, a, x)) == r'C_{n}^{\left(a\right)}\left(x\right)'
assert latex(gegenbauer(n, a, x)**2) == r'\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(chebyshevt(n, x)) == r'T_{n}\left(x\right)'
assert latex(
chebyshevt(n, x)**2) == r'\left(T_{n}\left(x\right)\right)^{2}'
assert latex(chebyshevu(n, x)) == r'U_{n}\left(x\right)'
assert latex(
chebyshevu(n, x)**2) == r'\left(U_{n}\left(x\right)\right)^{2}'
assert latex(legendre(n, x)) == r'P_{n}\left(x\right)'
assert latex(legendre(n, x)**2) == r'\left(P_{n}\left(x\right)\right)^{2}'
assert latex(
assoc_legendre(n, a, x)) == r'P_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_legendre(n, a, x)**2) == r'\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(laguerre(n, x)) == r'L_{n}\left(x\right)'
assert latex(laguerre(n, x)**2) == r'\left(L_{n}\left(x\right)\right)^{2}'
assert latex(
assoc_laguerre(n, a, x)) == r'L_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_laguerre(n, a, x)**2) == r'\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(hermite(n, x)) == r'H_{n}\left(x\right)'
assert latex(hermite(n, x)**2) == r'\left(H_{n}\left(x\right)\right)^{2}'
theta = Symbol("theta", real=True)
phi = Symbol("phi", real=True)
assert latex(Ynm(n,m,theta,phi)) == r'Y_{n}^{m}\left(\theta,\phi\right)'
assert latex(Ynm(n, m, theta, phi)**3) == r'\left(Y_{n}^{m}\left(\theta,\phi\right)\right)^{3}'
assert latex(Znm(n,m,theta,phi)) == r'Z_{n}^{m}\left(\theta,\phi\right)'
assert latex(Znm(n, m, theta, phi)**3) == r'\left(Z_{n}^{m}\left(\theta,\phi\right)\right)^{3}'
# Test latex printing of function names with "_"
assert latex(
polar_lift(0)) == r"\operatorname{polar\_lift}{\left (0 \right )}"
assert latex(polar_lift(
0)**3) == r"\operatorname{polar\_lift}^{3}{\left (0 \right )}"
assert latex(totient(n)) == r'\phi\left( n \right)'
# some unknown function name should get rendered with \operatorname
fjlkd = Function('fjlkd')
assert latex(fjlkd(x)) == r'\operatorname{fjlkd}{\left (x \right )}'
# even when it is referred to without an argument
assert latex(fjlkd) == r'\operatorname{fjlkd}'
# -*- coding: utf-8 -*-
"""
Created on Sun Sep 25 11:24:32 2016
@author: Lucas
"""
import lucaslib
import math
import sympy
answer = 0
for i in xrange(2, 1000001):
answer += sympy.totient(i)
print answer
#!/usr/bin/python3
# -*- coding: utf-8 -*-
# https://projecteuler.net/problem=70
from sympy import totient
LIMIT = 10**7
oran = 42
for n in range(2, LIMIT):
to = totient(n)
for p in set(str(n)):
if (str(n).count(p) != str(to).count(p)):
break
else:
if (set(str(n)) == set(str(to))):
if (len(set(str(n))) == len(set(str(to)))):
if (n / to < oran):
oran = n / to
enkucuk = n
print(enkucuk)
__author__ = 'Prateek'
from coPrime import coPrime
from sympy import totient
from elementOrder import order
def generators(n):
list = []
for elt in coPrime(n):
if order(n, elt) == totient(n):
list.append(elt)
return list
if __author__ == 'Prateek':
n = 13
print "Number of generators for",13," are",totient(totient(n))
print generators(n)
def test_F9():
assert totient(1776) == 576
def generators(n):
list = []
for elt in coPrime(n):
if order(n, elt) == totient(n):
list.append(elt)
return list
import sympy
from sympy import totient
from sympy import reduced_totient
import time
start = time.time()
N = 2019 #показатель
a = 2019 #основание
z = 10**2019
tot = totient(z)
i = 0
aa = a
f = open('2019.txt', 'w')
for i in range(N - 1):
r = aa % tot
f. write("r = "+str(r) + "\n")
print("r = "+str(r))
aa = pow(a,r,z)
f. write("a = "+str(aa) + "\n")
print("a = " + str(aa))
f. write("Ответ: "+str(aa) + "\n")
print(aa)
print(- start + time.time())
f. write("Время: "+str(- start + time.time()) + "\n")
def phi(n):
"""Calculates the value of phi(n), the number of positive integers
less than n and relatively prime to n."""
return sympy.totient(n)
