Find all non-empty finite sets of integers A and B with the following properties:
(i) Whenever x∈A, x+1∈B.
(ii) Whenever x∈B, x2-4∈A.
This is what I have found so far. This may be a complete list.
A B
[-3, 0] [-2, 1]
[-3, 0] [-2, 1, 2]
[-3, 0] [-2, -1, 1]
[-3, 0] [-2, -1, 1, 2]
[-3, 0, 1] [-2, 1, 2]
[-3, 0, 1] [-2, -1, 1, 2]
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I wrote a program where I manually change A and B and then see what comes out of it.
A = [0, 1]
B = []
cycles = 10
for c in range(cycles):
for a in A:
B.append(a+1)
for b in B:
A.append(b**2 - 4)
A = sorted(set(A))
B = sorted(set(B))
print(A,B, '\n')
Most inputs result in an ever increasing output. The few that don't, so far, stabilize in one of the above 6 sets.
An interesting finding:
Possibly relevant, I looked at the rules in the definition as being similar to two simultaneous equations:
v = u + 1
u = v^2 - 4
v - 1 = v^2 - 4
v^2 - v - 3 = 0
v = ( 1 ± √13)/2 about -1.3 and +2.3 like B
u = (-1 ± √13)/2 about -2.3 and +1.3 like A
Somewhat similar to the sets that were found.
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Posted by Larry
on 2024-06-19 10:22:13 |
Found so far
