What is the minimum number of members in a number group you can comfortably have before you can say that you have a pattern?
Hello, wonderboy1953!
The question is unanswerable . . . on several levels.What is the minimum number of members in a number group
you can comfortably have before you can say that you have a pattern?
Obviously, two terms is not enough.
. . $\displaystyle 1,\:2,\:\hdots$ could be the start of: .$\displaystyle \begin{Bmatrix}1,\:2,\:3,\:4,\:\hdots && f(n) \;=\; n \\
1,\:2,\:4,\:7,\:\hdots && f(n) \;=\; \frac{1}{2}(n^2-n+2) \end{Bmatrix}$
And three terms is insufficient.
. . $\displaystyle 1,\:2,\:4,\:\hdots $ could be the start of: .$\displaystyle \begin{Bmatrix}1,\:2,\:4,\:7,\:\hdots && f(n) \;=\; \frac{1}{2}(n^2-n+2) \\
1,\:2,\:4,\:8,\:\hdots && f(n) \;=\; 2^{n-1} \end{Bmatrix}$
Perhaps four terms is enough?
With: .$\displaystyle 1,\:2,\:4,\:8,\;\hdots$, we may suspect that we have a "doubling function",
. . and that the next term is: .$\displaystyle f(5)\,=\,16.$
But this sequence could be generated by: .$\displaystyle f(n) \:=\:(n-1)(n-2)(n-3)(n-4) + 2^{n-1}$
. . Hence: .$\displaystyle f(5) \,=\,40.$
Given any $\displaystyle n$ terms: .$\displaystyle a_1,\:a_2,\:a_3,\:\hdots \:a_n$
we can construct a polynomial function of degree $\displaystyle n\!-\!1$ which generates the $\displaystyle n$ values
and which can take on any value for the next term, $\displaystyle a_{n+1}$
Theoretically, "What is the next term?" problems are meaningless.
Yet if they "play fair", they are excellent exercises in Observation.
So we trust that: .$\displaystyle 1,\:4,\:9,\:16,\;\hdots$ are consecutive squares, $\displaystyle n^2$
. . . . . and that: .$\displaystyle 1,\:3,\:6,\:10,\:\hdots$ are consecutive triangular numbers, $\displaystyle \tfrac{1}{2}n(n+1)$
Obviously you're right and maybe I should have posted into the philosophy forum.
What I'm trying to find out is what the math community would find acceptable, at least at the conjecture stage, a proposition that looks likely to be true (e.g. Fermat's proposition before it became a theorem or Poincare's conjecture that was proven by Perelman).
I believe somewhere on Wolfram's site is a section where you can submit proposals for evaluation. Can someone direct me further?
I think it depends on what you mean by "group". Philosophically, I might argue that you need one more point than is minimally required to define the group. This presupposes that one can start by defining the relationship the numbers are meant to have. I think this is reasonable, because if you can't stipulate that up front, then you would never know if the pattern you found in a set of data was actually a complete description of the 'group' or just a description of a subset of the group.
For example, if two points define a line, then we need three points obeying the definition of that line (having only two would be a trivial solution) to say the numbers belong to that "group"...
There is no set amount that can determine a pattern.
How do we figure out if $\displaystyle a_n=n$?
If we are given 1,2,3.. This is surly not enough.
Say we are given 1,2,3,4,5...not enough
Say we are given 1,2,3,4,5.........,1773565 not enough.
For all normal purposes the answer is most defininantly $\displaystyle a_n=n$ but this does not mean that the 17777777788888888th digit is not 17777777788888890.
Just get a reasonable amount of data for the experiment (I would take 100-200 if it was something important and complex and if that amount of data was available). If it is not important but is complex, and data is available 20-30 should suffice.
Once you have proven the pattern for 100-200 possibilities, check 5 or 6 absurd numbers, something extremely large.
If not (such as the examples in the first reply) complex 5-10 is good, even for important things.
If sufficient data is not available, get as much as you can.