Number Theory ============= Sage has extensive functionality for number theory. For example, we can do arithmetic in :math:\ZZ/N\ZZ as follows: :: sage: R = IntegerModRing(97) sage: a = R(2) / R(3) sage: a 33 sage: a.rational_reconstruction() 2/3 sage: b = R(47) sage: b^20052005 50 sage: b.modulus() 97 sage: b.is_square() True Sage contains standard number theoretic functions. For example, :: sage: gcd(515,2005) 5 sage: factor(2005) 5 * 401 sage: c = factorial(25); c 15511210043330985984000000 sage: [valuation(c,p) for p in prime_range(2,23)] [22, 10, 6, 3, 2, 1, 1, 1] sage: next_prime(2005) 2011 sage: previous_prime(2005) 2003 sage: divisors(28); sum(divisors(28)); 2*28 [1, 2, 4, 7, 14, 28] 56 56 Perfect! Sage's sigma(n,k) function adds up the :math:k^{th} powers of the divisors of n: :: sage: sigma(28,0); sigma(28,1); sigma(28,2) 6 56 1050 We next illustrate the extended Euclidean algorithm, Euler's :math:\phi-function, and the Chinese remainder theorem: :: sage: d,u,v = xgcd(12,15) sage: d == u*12 + v*15 True sage: n = 2005 sage: inverse_mod(3,n) 1337 sage: 3 * 1337 4011 sage: prime_divisors(n) [5, 401] sage: phi = n*prod([1 - 1/p for p in prime_divisors(n)]); phi 1600 sage: euler_phi(n) 1600 sage: prime_to_m_part(n, 5) 401 We next verify something about the :math:3n+1 problem. :: sage: n = 2005 sage: for i in range(1000): ....: n = 3*odd_part(n) + 1 ....: if odd_part(n)==1: ....: print i ....: break 38 Finally we illustrate the Chinese remainder theorem. :: sage: x = crt(2, 1, 3, 5); x 11 sage: x % 3 # x mod 3 = 2 2 sage: x % 5 # x mod 5 = 1 1 sage: [binomial(13,m) for m in range(14)] [1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1] sage: [binomial(13,m)%2 for m in range(14)] [1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1] sage: [kronecker(m,13) for m in range(1,13)] [1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1] sage: n = 10000; sum([moebius(m) for m in range(1,n)]) -23 sage: Partitions(4).list() [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]] :math:p-adic Numbers ------------------------ The field of :math:p-adic numbers is implemented in Sage. Note that once a :math:p-adic field is created, you cannot change its precision. :: sage: K = Qp(11); K 11-adic Field with capped relative precision 20 sage: a = K(211/17); a 4 + 4*11 + 11^2 + 7*11^3 + 9*11^5 + 5*11^6 + 4*11^7 + 8*11^8 + 7*11^9 + 9*11^10 + 3*11^11 + 10*11^12 + 11^13 + 5*11^14 + 6*11^15 + 2*11^16 + 3*11^17 + 11^18 + 7*11^19 + O(11^20) sage: b = K(3211/11^2); b 10*11^-2 + 5*11^-1 + 4 + 2*11 + O(11^18) Much work has been done implementing rings of integers in :math:p-adic fields or number fields other than . The interested reader is invited to ask the experts on the sage-support Google group for further details. A number of related methods are already implemented in the NumberField class. :: sage: R. = PolynomialRing(QQ) sage: K = NumberField(x^3 + x^2 - 2*x + 8, 'a') sage: K.integral_basis() [1, 1/2*a^2 + 1/2*a, a^2] .. link :: sage: K.galois_group(type="pari") Galois group PARI group [6, -1, 2, "S3"] of degree 3 of the Number Field in a with defining polynomial x^3 + x^2 - 2*x + 8 .. link :: sage: K.polynomial_quotient_ring() Univariate Quotient Polynomial Ring in a over Rational Field with modulus x^3 + x^2 - 2*x + 8 sage: K.units() (3*a^2 + 13*a + 13,) sage: K.discriminant() -503 sage: K.class_group() Class group of order 1 of Number Field in a with defining polynomial x^3 + x^2 - 2*x + 8 sage: K.class_number() 1