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Some Problems in Computer Science and Elementary
Number Theory
Elwyn Berlekamp
Among most important unsolved problems in mathematics/ computer science
Does P = NP ?
Does there exist a polynomial time algorithm to solve the Traveling Salesman Problem?
=
The Traveling Salesman Problem
Given a graph (with n nodes), find a path which runs through all the nodes without any repeats.
Does this graph have a Hamiltonian Path?
NO(Proof coming later)
What about this graph?
YES
The Traveling Salesman Problem (All P- equivalent)
Version 1: Given a graph (with n nodes), find a path which runs through all the nodes without any repeats.
Version 1′: Determine whether or not such a path exists.
Version 2: Same as 1, except starting and ending points are given.Version 3: Given a graph, find a Hamiltonian cycle which runs through each node once.
Version 4: Given the complete graph of n nodes, and a table that specifies a cost to each of its n(n-1)/2 branches. Find the Hamiltonian cycle with least cost.
Version 5: Given a set of n integers: N={a1, a2, a3…an} and a set of pair sums; SS = {s1, s2, ...sk}, find a Hamiltonian path for the graph G whose nodes are NN, and there is a branch between ai and aj iff ai + aj ε S.
Interesting Special Case of the Traveling Salesman Problem:
Nodes = interval of j + 1- i consecutive integers: [ i , j ]
Permissible pairsums= SS = {s1, s2…}
We say [ i , j ] can be chained by SS iff a Hamiltonian path exist.
16
20
9
5
7
11
2
14
18
23
22
13
3
24
1
12 4
8 17
21 15
19
10
6
36
25 16
25 1625
9
25 36
25
25
36
16 16
25
4
9 25
99
3625
25
36
16
16 25 36
16
25
Problems: (wide range of difficulty)
For what value of n can[1, n] be chained by squares?
by cubes? by kth powers?
What is the smallest n such that[1, n] can be chained by squares?
…?
Is there a largest n such that[1, n] cannot be chained by squares?
…?If so, what is it?
S= {1, 4, 9, 16, 25, 36, 49, …}
1
2
3
4 5
6
7
8
9
10
111213
1
2
3
4 5
6
7
8
9
10
1112
13
14
S= {1, 4, 9, 16, 25, 36, 49, …}
1
2
3
4 5
6
7
8
9
10
1112
13
14
15
S= {1, 4, 9, 16, 25, 36, 49, …}
16
17
18
19
20
16
20
9
5
7
11
2
14
18
23 13
3
24
1
12 4
8 17
21 15
19
10
6
•If branch 2-14 is not used, then use of 18-7 forces an endpoint at 2 or 9.•If branch 2-14 is used, then there is an endpoint at 11 or 22.•So one endpoint is at 18; the other is among {2,9,11,22}•Branch 4-5 third endpoint at 20 or 11•Branch 3-6 third endpoint at 10 or 19•Branch 1-15 third endpoint at 21 or 10
Note: these reductions also work if nodes 24 and/or 23 are absent
9 2
11 22
Let’s now prove this graph has no Hamiltonian Path:
22
16
20 5
7
14
18
23 13
3
24
1
12 4
8 17
21 15
19
10
6
•Since 8 cannot be an endpoint, branch 1-8 must be used.
•Since 4 cannot be an endpoint, branch 12-4 must be used
•Since 24 cannot be an endpoint, branches 12-24 and 24-1 must be used
•But now [24,1,8,17,19,6,10,15,21,4,12] is a disjoint cycle
•So [1,24] cannot be chained by squares, QED
9 2
11 22
16
20
9
5
7
11
2
14
18
23
22
13
3
24
1
12 4
8 17
21 15
19
10
6
Can [1,22] be chained by squares?
16
20
9
5
7
11
2
14
18
22
13
3
1
12 4
8 17
21 15
19
10
6
NO
Can [1,22] be chained by squares?
16
20 5
7
14
18
23 13
3
1
12 4
8 17
21 15
19
10
6
•In [1,23], branch 13-3 would force a third endpoint at 12 or 23.
So it cannot be used.
9 2
11 22
What are all solutions of chaining [1,23] by squares?
16
20
9
5
7
11
2
14
18
23
22
13
3
1
12 4
8 17
21 15
19
10
6
•[1,23] can be chained by squares in exactly three different ways, with endpoints {18,9}, {18,2}, or {18,22}. Dotted lines cannot be used.
What are all solutions of chaining [1,23] by squares?
16
20
9
5
7
11
2
14
18
23
22
13
3
1
12 4
8 17
21 15
19
10
625 16
25 1625
25 36
25
25
3625
4
9 25 3625
16 25 36
16
25
[1,23] chained by squares
Conclusions:
[1,22] cannot be chained by squares
[1,23] CAN be chained by squares
[1,24] cannot be chained by squares
8, 1, 15, 10, 6, 3, 13, 12, 4, 5, 11, 14, 2, 7, 9
17,
, 16
Squares can chain [1,n] for n= 15, 16, and 17
And 23:
18, 7, 9, 16, 20, 5, 11, 14, 22, 3, 1, 8, 17, 19, 6, 10, 15, 21, 4, 12, 13, 23, 2.
And 25:18, 7, 9, 16, 20, 5, 11, 25, 24, 12, 4, 21, 15, 10, 6, 19, 17, 8, 1, 3,
22, 14, 2, 23, 13.
And 26:18, 7, 9, 16, 20, 5, 11, 25, 24, 12, 13, 3, 22, 14, 2, 23, 26, 10, 6, 19,
17, 8, 1, 15, 21, 4.
And 27:18, 7, 2, 14, 22, 27, 9, 16, 20, 5, 11, 25, 24, 12, 4, 21, 15, 10, 26, 23,
13, 3, 1, 8, 17, 19,6.
And 28:18, 7; 2, 23, 26, 10, 6, 19, 17, 8, 28, 21, 15, 1, 24, 25, 11; 14, 22,
27, 9, 16, 20; 5, 4, 12, 13, 3.
And 29:18, 7, (29), 20, 16, 9, 27, 22, 14; 2, 23, 26, 10, 6, 19, 17, 8, 28, 21, 15, 1, 24, 25, 11; 5, 4, 12, 13, 3.
And (now trivially) 30 and 31:
(31), 18, 7, 29, 20, 16, 9, 27, 22, 14, 2, 23, 26, 10; 6, (30), 19, 17, 8, 28, 21; 15, 1, 24, 25, 11, 5; 4, 12, because {6,19, 30} is the
first triangle in the infinite graph.
Here is another solution of 29, 30, and 31:
(31), 18, 7, 29, 20, 16, 9, 27, 22, 14, 2, 23, 26, 10; 15, 1, 3, 6, (30), 19, 17, 8, 28, 21, 4;
5, 11, 25, 24, 12, 13which extends to a solution of 31 and 32:
13, 12, 24, 25, 11, 5;
31, 18, 7, 29, 20, 16, 9, 27, 22, 14, 2, 23, 26, 10, 15, 1, 3, 6, 30, 19, 17, 8, 28, 21, 4, (32).
Problems: (wide range of difficulty)
For what value of n can[1, n] be chained by squares?
by cubes? by kth powers?
What is the smallest n such that[1, n] can be chained by squares?
…?
Is there a largest n such that[1, n] cannot be chained by squares?
…?If so, what is it?
[Vague?] How fast can the elements of S grow such that questions about chaining [1, n] remain interesting?
[RKG’s Conjecture] Fibonacci numbers, FF grow exponentially as
fast as any interesting set SS.
9 4
7
1
6
2 3
10
8
5
RKG:
FF chains [1, n] for n =
FF doesn’t chain [1, n] if n =
5 313
813
8
5 813
13
2, 3,4, 5,
6,
7,8,9,
10
11,
1113
21
12,
13
1312 2121 13
Fibonacci #Fibonacci # = {1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…}
Fibonacci plays Billiards!Joint unpublished result of ERB and RKG [2003]:
[1, Fk] is chained by {Fk-1, Fk, Fk+1}
Fibonacci plays Pool!
[1,34] is chained by {21,34,55}
Joint unpublished result of ERB and RKG [2003]:
[1, Fk] is chained by {Fk-1, Fk, Fk+1}
Fibonacci plays Pool!
[1,34] is chained by {21,34,55}
Pythagoras plays Billiards, Too!
If a, b, c, is a primitive Pythagorean triplet, with a <b <c and a²=b²=c², then [1, b²] is chained by squares
n = 15 is the smallest n such that [1, n] is chained by squares
If n < 23 and [1, n] is chained by squares, then it is chained by squares without using 2² = 4
†Small elements of SS aren’t of much use
Conditions for 4 elements of SS to form the corners of a billiard table:
B
A
C
D
A, B, C, D ε SS. . (A > B > C > D)
Corners are at A/2, B/2, C/2, D/2
Perimeter = n = A – C = B – D
Height = B – A = C – D
Width = B – C
Conditions for 4 elements of SS to form the corners of a billiard table:
B
A
C
D
A, B, C, D ε SS. . (A > B > C > D)
Corners are at A/2, B/2, C/2, D/2
Perimeter = n = A – C = B – D
Height = B – A = C – D
Width = B – C
If all corners are integers and if gcd(height, width) > 2, then path is degenerate.
If this gcd = 1, path is complete
If S = {s1 , s2 , …sk , …}
Where s1 < s2 < … < sk-1 < sk < …
And if sk + 2 ≤ n < sk+2 – (sk+2 )
Then S cannot chain [1, n]
Proof:
Corollaries: Fibs cannot chain [1, n] unless Fk – 2 ≤ n ≤ Fk + 1
Squares cannot chain [1, n] unless n ≥ 15
Cubes cannot chain [1, n] unless n ≥ 295
1 sk
sk+ 1sk+ 2
n
x = sk
y = x + 1
z = x + 2
FF chains [1, n] if n ε FF
FF chains [1, n] if n ε FF - 1
FF cannot chain [1, n] if FFk-1 + 1 < n < FFk - 1
Theorem
FF chains only 9 ε FF + 1
and only 11 ε FF - 2
127
216 89
7217233
161
377 233
144
51 21
17455
38
89 55
34
51 21
17455
38
Fk+2 Fk+1
Fk
3Fk
2
Fk+1
Fk-1
Fk+1 - Fk
2
Fk-1 - Fk
2 Fk
2
9
4
12
1
If sk+2 > sk+1 + sk + 1
and {s1, s2 , …, sk+2} chains [1,n]
then so does {s1, s2 , …, sk+1}
What is the fastest growing sequence such that for all k, there exists n(k), such that {s1, s2 , …, sk} chains [1, n]
but {s1, s2 , …, sk-1} does not?
Answer: Super- Fibonaccis: xn = xn-1 +xn-2 + 1
0, 1, 1, 3, 5, 9, 15, 25, 41, 68…
9
25 15
Engineering of Modified Pool Tables
0 1 8 27 64 216 343 7291000
1000
999 992 973 936 875 784 657 488 271 0
728 721 702 665 604 513 386 0
512 511 504 485 448 387 296 169 0
342 335 316 279 127 0
216 215 208 189 152 91 0
125 124 117 98 61 0
64 63 56 37 0
27 26 19 0
8 7 0
1 0
0
218218
217217
343343
729729
125125 512512
Can we make a useful pool table whose corners are CUBES?
343 125
512 729