2.10 Number Theory

Sage has extensive functionality for number theory. For example, we can do arithmetic in Z/NZ 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 kth 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 φ-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 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: list(partitions(4))
[(1, 1, 1, 1), (1, 1, 2), (2, 2), (1, 3), (4,)]



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