Sharedsupport / 2017.sagewsOpen in CoCalc
Author: William A. Stein
License: GNU General Public License v3.0
Description: Examples for support purposes.

http://weijr-note.blogspot.com/2017/01/2017-is-not-just-another-prime-number.html

is_prime(2017)
True
# The sum of all odd primes up to 2017 is a prime number, i.e., # 3+5+7+11+...+2017 is a prime number. n = sum(primes(3,2018)); n is_prime(n)
283079 True
# 2017π (rounds to nearest integer) is a prime n = round(2017*pi); n is_prime(n)
6337 True
# 2017e (rounds to nearest integer ) is a prime. n = round(2017*e); n is_prime(n)
5483 True
# The sum of the cubes of gaps of primes up to 2017 is a prime number. v = prime_range(2018) n = sum([(v[i+1]-v[i])^3 for i in range(len(v)-1)]); n is_prime(n)
258569 True
# The prime number before 2017 is 2017+(2-0-1-7), which makes it a # sexy prime, and the prime after 2017 is 2017+(2+0+1+7). 2017 itself # is of course equal to 2017+(2*0*1*7) 2017+(2-0-1-7) == previous_prime(2017) 2017+(2+0+1+7) == next_prime(2017)
True True
# Insert 7 into any two digits of 2017, it is still a prime number, # i.e. 27017, 20717, 20177 are all primes. for n in [27017, 20717, 20177]: is_prime(n)
True True True
# Since all digits of 2017 is less than 8, it can be viewed as an octal. # 2017 is still a prime number as an octal. n = int('2017',8); n is_prime(n)
1039 True
# 2017 can be written as a sum of three cubes of primes, i.e., # p^3 +q^3 +r^3 for some primes p, q, r k = int(2017^(1/3)) for p in [1..k]: for q in [1..k]: for r in [1..k]: if p^3 + q^3 + r^3 == 2017: print p, q, r 7^3 + 7^3 + 11^3 == 2017
# can be written as a sum of cubes of five distinct integers. k = int(2017^(1/3)) n = 0 for p in [1..k]: for q in [1..k]: for r in [1..k]: for s in [1..k]: for t in [1..k]: if p^3 + q^3 + r^3 + s^3 + t^3 == 2017: n += 1 print "%s ways"%n
# 2017 can be written as # x^2+y^2, x^2+2y^2, x^2+3y^2, x^2+4y^2 x^2+6y^2, x^2+7y^2, x^2+8y^2, x^2+9y^2 # (for positive integers x, y) R.<x,y> =ZZ[] B = int(sqrt(2017)) for f in [x^2+y^2, x^2+2*y^2, x^2+3*y^2, x^2+4*y^2, x^2+6*y^2, x^2+7*y^2, x^2+8*y^2, x^2+9*y^2]: done = False for a in [1..B]: for b in [1..B]: if f(a,b) == 2017: print str(f).replace('x', str(a)).replace('y', str(b)) + ' == 2017' done = True break if done: break
9^2 + 44^2 == 2017 37^2 + 2*18^2 == 2017 17^2 + 3*24^2 == 2017 9^2 + 4*22^2 == 2017 29^2 + 6*14^2 == 2017 15^2 + 7*16^2 == 2017 37^2 + 8*9^2 == 2017 44^2 + 9*3^2 == 2017
# 20170123456789 is also a prime is_prime(20170123456789)
True
# the 2017th prime number is 17539 and 201717539 is also a prime. nth_prime(2017) is_prime(201717539)
17539 True
# Let p=2017, then both (p+1)/2 and (p+2)/3 are prime numbers. p = 2017 is_prime((p+1)//2) is_prime((p+2)//3)
True True
# The first ten digits of the decimal expansion of the cubic root of # 2017 contains all different digits 0~9. N(2017^(1/3),digits=15) len(set(str(N(2017^(1/3),digits=15)).replace('.','')[:10])) # 2017 is the least integer that has this property. print "check who has this property" for n in [1..2017]: if len(set(str(N(n^(1/3),digits=15)).replace('.','')[:10])) == 10: print "n =", n, " has this property"
12.6348075933001 10 check who has this property n = 2017 has this property
N(10^(1/3), digits=15) len(set(str(N(10^(1/3),digits=15)).replace('.','')[:10]))
2.15443469003188 8
# 2017 = 2^11 - 11th prime 2017 == 2^11 - nth_prime(11)
True

Plus stuff I care about...

sum_of_k_squares(2, 2017)
(9, 44)
E = EllipticCurve('2017a'); show(E)
y2+xy=x3+x210x+9\displaystyle y^2 + x y = x^{3} + x^{2} - 10 x + 9
E.rank()
1
ModularSymbols(2017, 2,sign=1).decomposition()
[ Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 168 for Gamma_0(2017) of weight 2 with sign 1 over Rational Field, Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 168 for Gamma_0(2017) of weight 2 with sign 1 over Rational Field, Modular Symbols subspace of dimension 80 of Modular Symbols space of dimension 168 for Gamma_0(2017) of weight 2 with sign 1 over Rational Field, Modular Symbols subspace of dimension 86 of Modular Symbols space of dimension 168 for Gamma_0(2017) of weight 2 with sign 1 over Rational Field ]