Let us investigate the 3n+1 problem. We'll define f and g as described in the 2023-02-03 notes.
def f(n):
if is_even(n):
return(n/2)
else:
return(3*n+1)
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f(2)
1 1 |
type(f(2))
<type 'sage.rings.rational.Rational'> <type 'sage.rings.rational.Rational'> |
type(Integer(f(2)))
<type 'sage.rings.integer.Integer'> <type 'sage.rings.integer.Integer'> |
Integer(f(2))
1 1 |
Integer(3/4)
Traceback (click to the left of this block for traceback) ... TypeError: no conversion of this rational to integer Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_8.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("SW50ZWdlcigzLzQp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmpxmdAUC/___code___.py", line 3, in <module>
exec compile(u'Integer(_sage_const_3 /_sage_const_4 )
File "", line 1, in <module>
File "sage/rings/integer.pyx", line 662, in sage.rings.integer.Integer.__init__ (/usr/local/sage-6.10/src/build/cythonized/sage/rings/integer.c:5806)
File "sage/rings/rational.pyx", line 2670, in sage.rings.rational.Rational._integer_ (/usr/local/sage-6.10/src/build/cythonized/sage/rings/rational.c:23379)
TypeError: no conversion of this rational to integer
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Let us improve the function f: we'll check that the input is an integer, and is positive.
def f(n):
if n<1 or (type(n)==Integer)==False:
return("Please input a positive integer")
elif is_even(n):
return(Integer(n/2))
else:
return(3*n+1)
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f(3)
10 10 |
type(f(3))
<type 'sage.rings.integer.Integer'> <type 'sage.rings.integer.Integer'> |
type(f(4))
<type 'sage.rings.integer.Integer'> <type 'sage.rings.integer.Integer'> |
Let us also define g, which computes (3n+1)/2 in one step.
def g(n):
if n<1 or (type(n)==Integer)==False:
return("Please input a positive integer")
elif is_even(n):
return(Integer(n/2))
else:
return(Integer((3*n+1)/2))
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g(3)
5 5 |
type(g(3))
<type 'sage.rings.integer.Integer'> <type 'sage.rings.integer.Integer'> |
f(2)
1 1 |
f(7)
22 22 |
f(22)
11 11 |
f(11)
34 34 |
f(34)
17 17 |
f(17)
52 52 |
f(52)
26 26 |
f(26)
13 13 |
f(13)
40 40 |
f(40)
20 20 |
f(20)
10 10 |
f(10)
5 5 |
f(5)
16 16 |
f(16)
8 8 |
Since we believe we will always reach 1, let's create a while loop to compute how many steps for f to reach 1.
step_counter = 0
n=7
while n>1:
step_counter+=1
n=f(n)
print(step_counter)
16 16 |
step_counter = 0
n=9
while n>1:
step_counter+=1
n=f(n)
print(step_counter)
19 19 |
Let us think about creating a list of pairs [n,hs(n) ] where hs(n) is the number of steps needed to reach 1.
hailstone=[ [1,0] ]
for m in srange(2,20):
step_counter=0
n=m
while n>1:
step_counter+=1
n=f(n)
hailstone.append([m,step_counter])
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print(hailstone)
[[1, 0], [2, 1], [3, 7], [4, 2], [5, 5], [6, 8], [7, 16], [8, 3], [9, 19], [10, 6], [11, 14], [12, 9], [13, 9], [14, 17], [15, 17], [16, 4], [17, 12], [18, 20], [19, 20]] [[1, 0], [2, 1], [3, 7], [4, 2], [5, 5], [6, 8], [7, 16], [8, 3], [9, 19], [10, 6], [11, 14], [12, 9], [13, 9], [14, 17], [15, 17], [16, 4], [17, 12], [18, 20], [19, 20]] |
list_plot(hailstone)
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hailstone=[ [1,0] ]
for m in srange(2,1000):
step_counter=0
n=m
while n>1:
step_counter+=1
n=f(n)
hailstone.append([m,step_counter])
list_plot(hailstone)
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hailstone=[ [1,0] ]
for m in srange(2,10000):
step_counter=0
n=m
while n>1:
step_counter+=1
n=f(n)
hailstone.append([m,step_counter])
list_plot(hailstone)
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hailstone=[ [1,0] ]
for m in srange(2,100000):
step_counter=0
n=m
while n>1:
step_counter+=1
n=f(n)
hailstone.append([m,step_counter])
list_plot(hailstone)
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