We know that if $n \equiv 3 \pmod{4}$
def hailstone2(n):
if is_even(n):
return(Integer(n/2))
else:
return(Integer((3*n+1)/2))
|
|
def hailstone_below(n):
count=0
j=n
while j>=n:
j=hailstone2(j)
count+=1
#print(j)
return(count)
|
|
hailstone2(1)
2 2 |
hailstone_below(5)
2 2 |
hailstone_below_list=[]
for j in srange(1,100000):
hailstone_below_list.append([4*j+3,hailstone_below(4*j+3)])
|
|
list_plot(hailstone_below_list[:100])
|
it is apparent that there are regular values for which it takes exactly 4 steps to get below $n$.
for pairs in hailstone_below_list[:100]:
if pairs[1]==4:
print(pairs, mod(pairs[0],16))
([19, 4], 3) ([35, 4], 3) ([51, 4], 3) ([67, 4], 3) ([83, 4], 3) ([99, 4], 3) ([115, 4], 3) ([131, 4], 3) ([147, 4], 3) ([163, 4], 3) ([179, 4], 3) ([195, 4], 3) ([211, 4], 3) ([227, 4], 3) ([243, 4], 3) ([259, 4], 3) ([275, 4], 3) ([291, 4], 3) ([307, 4], 3) ([323, 4], 3) ([339, 4], 3) ([355, 4], 3) ([371, 4], 3) ([387, 4], 3) ([403, 4], 3) ([19, 4], 3) ([35, 4], 3) ([51, 4], 3) ([67, 4], 3) ([83, 4], 3) ([99, 4], 3) ([115, 4], 3) ([131, 4], 3) ([147, 4], 3) ([163, 4], 3) ([179, 4], 3) ([195, 4], 3) ([211, 4], 3) ([227, 4], 3) ([243, 4], 3) ([259, 4], 3) ([275, 4], 3) ([291, 4], 3) ([307, 4], 3) ([323, 4], 3) ([339, 4], 3) ([355, 4], 3) ([371, 4], 3) ([387, 4], 3) ([403, 4], 3) |
So, it appears that if $ n \equiv 3 \pmod{16}$ then we take exactly 4 steps to get below $n$.
for pairs in hailstone_below_list[:100]:
if pairs[1]==5:
print(pairs, mod(pairs[0],16))
([11, 5], 11) ([23, 5], 7) ([43, 5], 11) ([55, 5], 7) ([75, 5], 11) ([87, 5], 7) ([107, 5], 11) ([119, 5], 7) ([139, 5], 11) ([151, 5], 7) ([171, 5], 11) ([183, 5], 7) ([203, 5], 11) ([215, 5], 7) ([235, 5], 11) ([247, 5], 7) ([267, 5], 11) ([279, 5], 7) ([299, 5], 11) ([311, 5], 7) ([331, 5], 11) ([343, 5], 7) ([363, 5], 11) ([375, 5], 7) ([395, 5], 11) ([11, 5], 11) ([23, 5], 7) ([43, 5], 11) ([55, 5], 7) ([75, 5], 11) ([87, 5], 7) ([107, 5], 11) ([119, 5], 7) ([139, 5], 11) ([151, 5], 7) ([171, 5], 11) ([183, 5], 7) ([203, 5], 11) ([215, 5], 7) ([235, 5], 11) ([247, 5], 7) ([267, 5], 11) ([279, 5], 7) ([299, 5], 11) ([311, 5], 7) ([331, 5], 11) ([343, 5], 7) ([363, 5], 11) ([375, 5], 7) ([395, 5], 11) |
These values appear to differ by 32 each time, rather than by sixteen.
for pairs in hailstone_below_list[:100]:
if pairs[1]==5:
print(pairs, mod(pairs[0],32))
([11, 5], 11) ([23, 5], 23) ([43, 5], 11) ([55, 5], 23) ([75, 5], 11) ([87, 5], 23) ([107, 5], 11) ([119, 5], 23) ([139, 5], 11) ([151, 5], 23) ([171, 5], 11) ([183, 5], 23) ([203, 5], 11) ([215, 5], 23) ([235, 5], 11) ([247, 5], 23) ([267, 5], 11) ([279, 5], 23) ([299, 5], 11) ([311, 5], 23) ([331, 5], 11) ([343, 5], 23) ([363, 5], 11) ([375, 5], 23) ([395, 5], 11) ([11, 5], 11) ([23, 5], 23) ([43, 5], 11) ([55, 5], 23) ([75, 5], 11) ([87, 5], 23) ([107, 5], 11) ([119, 5], 23) ([139, 5], 11) ([151, 5], 23) ([171, 5], 11) ([183, 5], 23) ([203, 5], 11) ([215, 5], 23) ([235, 5], 11) ([247, 5], 23) ([267, 5], 11) ([279, 5], 23) ([299, 5], 11) ([311, 5], 23) ([331, 5], 11) ([343, 5], 23) ([363, 5], 11) ([375, 5], 23) ([395, 5], 11) |
So, there are two values $\pmod{32}$ which take exactly 5 steps to get below the starting value. Note that $32=2^5$!
for pairs in hailstone_below_list:
if pairs[1]==6:
print(pairs, mod(pairs[0],16))
|
|
So nothing takes exactly 6 steps to get below the starting value.
for pairs in hailstone_below_list[:100]:
if pairs[1]==7:
print(pairs, mod(pairs[0],128))
([7, 7], 7) ([15, 7], 15) ([59, 7], 59) ([135, 7], 7) ([143, 7], 15) ([187, 7], 59) ([263, 7], 7) ([271, 7], 15) ([315, 7], 59) ([391, 7], 7) ([399, 7], 15) ([7, 7], 7) ([15, 7], 15) ([59, 7], 59) ([135, 7], 7) ([143, 7], 15) ([187, 7], 59) ([263, 7], 7) ([271, 7], 15) ([315, 7], 59) ([391, 7], 7) ([399, 7], 15) |
There appear to be 3 congruences $\pmod{128}$ which take 7 steps to get below the initial value. Here again, 128 is the
correct modulus, and $128=2^7$!
for pairs in hailstone_below_list[:200]:
if pairs[1]==8:
print(pairs, mod(pairs[0],256))
([39, 8], 39) ([79, 8], 79) ([95, 8], 95) ([123, 8], 123) ([175, 8], 175) ([199, 8], 199) ([219, 8], 219) ([295, 8], 39) ([335, 8], 79) ([351, 8], 95) ([379, 8], 123) ([431, 8], 175) ([455, 8], 199) ([475, 8], 219) ([551, 8], 39) ([591, 8], 79) ([607, 8], 95) ([635, 8], 123) ([687, 8], 175) ([711, 8], 199) ([731, 8], 219) ([39, 8], 39) ([79, 8], 79) ([95, 8], 95) ([123, 8], 123) ([175, 8], 175) ([199, 8], 199) ([219, 8], 219) ([295, 8], 39) ([335, 8], 79) ([351, 8], 95) ([379, 8], 123) ([431, 8], 175) ([455, 8], 199) ([475, 8], 219) ([551, 8], 39) ([591, 8], 79) ([607, 8], 95) ([635, 8], 123) ([687, 8], 175) ([711, 8], 199) ([731, 8], 219) |
There are exactly 7 congruences $\pmod{256}$ which take exactly 8 steps to get below the initial value. 256 is the correct modulus, and $256=2^8$.
for pairs in hailstone_below_list:
if pairs[1]==9:
print(pairs, mod(pairs[0],512))
|
|
There's nothing for 9 steps
for pairs in hailstone_below_list[:800]:
if pairs[1]==10:
print(pairs, mod(pairs[0],1024))
([287, 10], 287) ([347, 10], 347) ([367, 10], 367) ([423, 10], 423) ([507, 10], 507) ([575, 10], 575) ([583, 10], 583) ([735, 10], 735) ([815, 10], 815) ([923, 10], 923) ([975, 10], 975) ([999, 10], 999) ([1311, 10], 287) ([1371, 10], 347) ([1391, 10], 367) ([1447, 10], 423) ([1531, 10], 507) ([1599, 10], 575) ([1607, 10], 583) ([1759, 10], 735) ([1839, 10], 815) ([1947, 10], 923) ([1999, 10], 975) ([2023, 10], 999) ([2335, 10], 287) ([2395, 10], 347) ([2415, 10], 367) ([2471, 10], 423) ([2555, 10], 507) ([2623, 10], 575) ([2631, 10], 583) ([2783, 10], 735) ([2863, 10], 815) ([2971, 10], 923) ([3023, 10], 975) ([3047, 10], 999) ([287, 10], 287) ([347, 10], 347) ([367, 10], 367) ([423, 10], 423) ([507, 10], 507) ([575, 10], 575) ([583, 10], 583) ([735, 10], 735) ([815, 10], 815) ([923, 10], 923) ([975, 10], 975) ([999, 10], 999) ([1311, 10], 287) ([1371, 10], 347) ([1391, 10], 367) ([1447, 10], 423) ([1531, 10], 507) ([1599, 10], 575) ([1607, 10], 583) ([1759, 10], 735) ([1839, 10], 815) ([1947, 10], 923) ([1999, 10], 975) ([2023, 10], 999) ([2335, 10], 287) ([2395, 10], 347) ([2415, 10], 367) ([2471, 10], 423) ([2555, 10], 507) ([2623, 10], 575) ([2631, 10], 583) ([2783, 10], 735) ([2863, 10], 815) ([2971, 10], 923) ([3023, 10], 975) ([3047, 10], 999) |
There are 12 congruences $\pmod{1024}$, 1024 is the right modulus, and $2^{10}=1024$, of course.
for pairs in hailstone_below_list:
if pairs[1]==11:
print(pairs, mod(pairs[0],1024))
|
|
for pairs in hailstone_below_list[:3000]:
if pairs[1]==12:
print(pairs, mod(pairs[0],4096))
([231, 12], 231) ([383, 12], 383) ([463, 12], 463) ([615, 12], 615) ([879, 12], 879) ([935, 12], 935) ([1019, 12], 1019) ([1087, 12], 1087) ([1231, 12], 1231) ([1435, 12], 1435) ([1647, 12], 1647) ([1703, 12], 1703) ([1787, 12], 1787) ([1823, 12], 1823) ([1855, 12], 1855) ([2031, 12], 2031) ([2203, 12], 2203) ([2239, 12], 2239) ([2351, 12], 2351) ([2587, 12], 2587) ([2591, 12], 2591) ([2907, 12], 2907) ([2975, 12], 2975) ([3119, 12], 3119) ([3143, 12], 3143) ([3295, 12], 3295) ([3559, 12], 3559) ([3675, 12], 3675) ([3911, 12], 3911) ([4063, 12], 4063) ([4327, 12], 231) ([4479, 12], 383) ([4559, 12], 463) ([4711, 12], 615) ([4975, 12], 879) ([5031, 12], 935) ([5115, 12], 1019) ([5183, 12], 1087) ([5327, 12], 1231) ([5531, 12], 1435) ([5743, 12], 1647) ([5799, 12], 1703) ([5883, 12], 1787) ([5919, 12], 1823) ([5951, 12], 1855) ([6127, 12], 2031) ([6299, 12], 2203) ([6335, 12], 2239) ([6447, 12], 2351) ([6683, 12], 2587) ([6687, 12], 2591) ([7003, 12], 2907) ([7071, 12], 2975) ([7215, 12], 3119) ([7239, 12], 3143) ([7391, 12], 3295) ([7655, 12], 3559) ([7771, 12], 3675) ([8007, 12], 3911) ([8159, 12], 4063) ([8423, 12], 231) ([8575, 12], 383) ([8655, 12], 463) ([8807, 12], 615) ([9071, 12], 879) ([9127, 12], 935) ([9211, 12], 1019) ([9279, 12], 1087) ([9423, 12], 1231) ([9627, 12], 1435) ([9839, 12], 1647) ([9895, 12], 1703) ([9979, 12], 1787) ([10015, 12], 1823) ([10047, 12], 1855) ([10223, 12], 2031) ([10395, 12], 2203) ([10431, 12], 2239) ([10543, 12], 2351) ([10779, 12], 2587) ([10783, 12], 2591) ([11099, 12], 2907) ([11167, 12], 2975) ([11311, 12], 3119) ([11335, 12], 3143) ([11487, 12], 3295) ([11751, 12], 3559) ([11867, 12], 3675) ([231, 12], 231) ([383, 12], 383) ([463, 12], 463) ([615, 12], 615) ([879, 12], 879) ([935, 12], 935) ([1019, 12], 1019) ([1087, 12], 1087) ([1231, 12], 1231) ([1435, 12], 1435) ([1647, 12], 1647) ([1703, 12], 1703) ([1787, 12], 1787) ([1823, 12], 1823) ([1855, 12], 1855) ([2031, 12], 2031) ([2203, 12], 2203) ([2239, 12], 2239) ([2351, 12], 2351) ([2587, 12], 2587) ([2591, 12], 2591) ([2907, 12], 2907) ([2975, 12], 2975) ([3119, 12], 3119) ([3143, 12], 3143) ([3295, 12], 3295) ([3559, 12], 3559) ([3675, 12], 3675) ([3911, 12], 3911) ([4063, 12], 4063) ([4327, 12], 231) ([4479, 12], 383) ([4559, 12], 463) ([4711, 12], 615) ([4975, 12], 879) ([5031, 12], 935) ([5115, 12], 1019) ([5183, 12], 1087) ([5327, 12], 1231) ([5531, 12], 1435) ([5743, 12], 1647) ([5799, 12], 1703) ([5883, 12], 1787) ([5919, 12], 1823) ([5951, 12], 1855) ([6127, 12], 2031) ([6299, 12], 2203) ([6335, 12], 2239) ([6447, 12], 2351) ([6683, 12], 2587) ([6687, 12], 2591) ([7003, 12], 2907) ([7071, 12], 2975) ([7215, 12], 3119) ([7239, 12], 3143) ([7391, 12], 3295) ([7655, 12], 3559) ([7771, 12], 3675) ([8007, 12], 3911) ([8159, 12], 4063) ([8423, 12], 231) ([8575, 12], 383) ([8655, 12], 463) ([8807, 12], 615) ([9071, 12], 879) ([9127, 12], 935) ([9211, 12], 1019) ([9279, 12], 1087) ([9423, 12], 1231) ([9627, 12], 1435) ([9839, 12], 1647) ([9895, 12], 1703) ([9979, 12], 1787) ([10015, 12], 1823) ([10047, 12], 1855) ([10223, 12], 2031) ([10395, 12], 2203) ([10431, 12], 2239) ([10543, 12], 2351) ([10779, 12], 2587) ([10783, 12], 2591) ([11099, 12], 2907) ([11167, 12], 2975) ([11311, 12], 3119) ([11335, 12], 3143) ([11487, 12], 3295) ([11751, 12], 3559) ([11867, 12], 3675) |
There are 30 congruences $\pmod{2^{12}}$ which require exactly 12 steps to get below the starting value.
m=11
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
print(pairs, mod(pairs[0],2^m))
paircount+=1
print paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
print(pairs, mod(pairs[0],2^(m+1)))
paircount+=1
print paircount
0 0 0 0 |
m=12
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
#print(pairs, mod(pairs[0],2^m))
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
#print(pairs, mod(pairs[0],2^(m+1)))
paircount+=1
print paircount
12 30 60 12 30 60 |
m=13
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
13 85 14 165 15 340 16 680 17 1360 13 85 14 165 15 340 16 680 17 1360 |
This suggests very strongly that for some of the numbers which take exactly 13 steps to get below the starting value, the correct modulus is $2^{15}$ rather than $2^{13}$.
m=14
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
14 0 15 0 16 0 17 0 18 0 14 0 15 0 16 0 17 0 18 0 |
m=15
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
15 88 16 88 17 692 18 1384 19 2768 15 88 16 88 17 692 18 1384 19 2768 |
Again, this suggests that something strange is going on with 15 steps. Notice that there are very few instances of 15 in the first $2^{16}$ values!
m=16
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
16 114 17 114 18 1904 19 3808 20 7616 16 114 17 114 18 1904 19 3808 20 7616 |
m=17
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
17 0 18 0 19 0 20 0 21 0 17 0 18 0 19 0 20 0 21 0 |
m=18
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
18 63 19 63 20 3844 21 7688 22 15376 18 63 19 63 20 3844 21 7688 22 15376 |
m=19
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
19 0 20 0 21 0 22 0 23 0 19 0 20 0 21 0 22 0 23 0 |
m=20
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
20 56 21 56 22 10608 23 21216 24 42432 20 56 21 56 22 10608 23 21216 24 42432 |
m=21
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
21 53 22 53 23 32180 24 64360 25 128720 21 53 22 53 23 32180 24 64360 25 128720 |
m=22
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
22 0 23 0 24 0 25 0 26 0 22 0 23 0 24 0 25 0 26 0 |
m=23
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
23 27 24 27 25 70548 26 84143 27 84143 23 27 24 27 25 70548 26 84143 27 84143 |
This just indicates we've gone beyond the end of hailstone_below_list
2^26
67108864 67108864 |
hailstone_below_list[-1]
[39999999, 21] [39999999, 21] |
Let's start again from the beginning.
If n is even, it takes 1 step.
If n is $1 \pmod{4}$ it takes 2 steps.
There are no congruences that take 3 steps.
There is 1 congruence $\pmod{2^4}$ that takes 4 steps.
There are 2 congruences $\pmod{2^5}$ that take 5 steps.
There are no congruences that take 6 steps.
There are 3 congruences $\pmod{2^7}$ that take 7 steps.
There are 7 congruences $\pmod{2^8}$ that take 8 steps.
There are no congruences that take 9 steps.
There are 12 congruences $\pmod{2^{10}}$ that take 10 steps.
There are no congruences that take 11 steps.
There are 30 congruences $\pmod{2^{12}}$ that take 12 steps.
There are 340 congruences $\pmod{2^{15}}$ that take 13 steps.
There are no congruences that take 14 steps.
There are 692 congruences $\pmod{2^{17}}$ that take 15 steps.
There are 1906 congruences $\pmod{2^{18}}$ that take 16 steps.
m=16
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
16 114 17 114 18 1904 19 3808 20 7616 16 114 17 114 18 1904 19 3808 20 7616 |
m=4
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
4 0 5 1 6 3 7 7 8 15 4 0 5 1 6 3 7 7 8 15 |
These are all off by 1 because I didn't include the value 3 in the list.
m=5
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
5 2 6 4 7 8 8 16 9 32 5 2 6 4 7 8 8 16 9 32 |
m=6
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
6 0 7 0 8 0 9 0 10 0 6 0 7 0 8 0 9 0 10 0 |
m=7
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
7 3 8 6 9 12 10 24 11 48 7 3 8 6 9 12 10 24 11 48 |
m=8
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
8 7 9 14 10 28 11 56 12 112 8 7 9 14 10 28 11 56 12 112 |
m=9
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
9 0 10 0 11 0 12 0 13 0 9 0 10 0 11 0 12 0 13 0 |
m=10
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
10 12 11 24 12 48 13 96 14 192 10 12 11 24 12 48 13 96 14 192 |
m=11
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
11 0 12 0 13 0 14 0 15 0 11 0 12 0 13 0 14 0 15 0 |
m=12
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
12 30 13 60 14 120 15 240 16 480 12 30 13 60 14 120 15 240 16 480 |
m=21
paircount = 0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^m:
break
if pairs[1]==m:
paircount+=1
print m,paircount
paircount=0
for pairs in hailstone_below_list[:4000]:
if pairs[0]>2^(m+1):
break
if pairs[1]==m:
paircount+=1
print m+1,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+2):
break
if pairs[1]==m:
paircount+=1
print m+2,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+3):
break
if pairs[1]==m:
paircount+=1
print m+3,paircount
paircount=0
for pairs in hailstone_below_list:
if pairs[0]>2^(m+4):
break
if pairs[1]==m:
paircount+=1
print m+4,paircount
21 53 22 53 23 32180 24 64360 25 128720 21 53 22 53 23 32180 24 64360 25 128720 |
prob= 1/2 + 1/4+1/2^4 + 2/2^5 + 3/2^7 + 7/2^8 + 12/2^10 + 30/2^12 + 340/2^15 + 692/2^17 + 1906/2^18 + 3844/2^20 + 10608/2^22 + 32180/2^23
|
|
prob.n()
0.977781772613525 0.977781772613525 |
N=10000000
stepprob=[]
for steps in range(2,100):
tempprob=0
for pair in hailstone_below_list[:N]:
#print pair
if pair[1]<=steps:
tempprob+=1
tempprob=(tempprob/N).n()
stepprob.append([steps,tempprob])
#print stepprob
|
|
stepprob[-1]
[99, 0.998953300000000] [99, 0.998953300000000] |
y=2000
N=2^y # Range of integers from which we'll choose a random number
R=10000 # number of random trials
L=1000 # largest step we'll track
max_seen=0 # track the largest number of steps we see
distn=[0,0,0,0,0,0,0,0] # keep track of how many we see (mod 8)
steps_seen=[0 for j in srange(L)]
for j in srange(R):
n=ZZ.random_element(x=N)
distn[mod(n,8)]+=1
steps_taken=hailstone_below(n)
if steps_taken > max_seen:
max_seen=steps_taken
#print steps_taken,is_even(n)
if steps_taken<L:
steps_seen[steps_taken]+=1
|
|
max_seen
132 132 |
print steps_seen[:20]
print distn
[0, 5091, 2466, 0, 643, 628, 0, 221, 250, 0, 125, 0, 58, 97, 0, 44, 75, 0, 38, 0] [1262, 1223, 1280, 1251, 1277, 1243, 1272, 1192] [0, 5091, 2466, 0, 643, 628, 0, 221, 250, 0, 125, 0, 58, 97, 0, 44, 75, 0, 38, 0] [1262, 1223, 1280, 1251, 1277, 1243, 1272, 1192] |
list_plot(steps_seen[2:])
|
only_steps_seen=[]
for j in range(L):
if steps_seen[j]>0:
only_steps_seen.append([j,steps_seen[j]])
|
|
list_plot(only_steps_seen[30:])
|
max(steps_seen)
50143 50143 |
hailstone_below_list_singles=[]
for pair in hailstone_below_list:
hailstone_below_list_singles.append(pair[1])
|
|
max(hailstone_below_list_singles)
298 298 |
hailstone_below(17)
2 2 |
|
|