Recent content by erszega

D_H's f(g(x)) appears to provide solutions for x^y = y^x, x<>y, x>e. For instance y=f(g(3)) is the other solution for 3^y = y^3, besides the obvious y=3. When x=e, the only solution is y=e. However, such solutions also exist when 1 < x < e, eg 2^4 = 4^2, but f(g(x)) does not help then, eg...

Apologies, I meant the frequency of each S(i) or P(i). It may be better to describe what I did in the following way: I used a spreadsheet, and created a table, consisting of, say, 10,000 rows and 5 columns, of random numbers (using the spreadsheet's random number function). Then I added (or...

Is it correct to say that independent random events (additively) lead to a normal distribution, and dependent random events (multiplicatively) lead to a power law distribution? The following might be trivial, but it was quite interesting to find for me, someone with a very limited knowledge...

This property seems to hold not only for cycles but for any sequence of such iterations if they are long enough without repititive cycles. Something similar as a power law also appears when the size of the numbers and their frequency is analysed. As an example, I looked at 65535 numbers in a...

I am not sure if this can lead to any insights, but there seems to be something like a "power law" behind these cycles. For instance, taking the cycle that F(n)=sopfr(F(n-1)+F(n-2)+F(n-3)+F(n-4)+F(n-5)) (often or always??) leads to (cycle length: 31196, take initial values 49,93,435,98,92, for...

Dear All, Could you give me examples of conjectures based on perceived patterns but proved to be wrong? Fermat numbers, with Fermat's conjecture that all Fermat numbers are primes, would be one example that I know of. I would appreciate elementary examples which are easy to understand. The...

Just to say that these cycles are like "attractors". For instance, a sequence of type F(n) = sopfr(F(n-1)+F(n-2)) seems to get trapped in one of the four cycles I described in my first post, whatever the first two terms (although I admit I tested only a very small range). Eg: 20000, 20000...

The reason I posted this is just that I am curious to find out if this sort of iteration of the sopfr function necessarily leads to cycles. 1) I just wanted to provide examples. By the way, a very simple example: sopfr(8+8) = 8, and so the cycle length is simply 1. Another example...

I posted this in another math forum, but have received no response: Let sopfr(n) be the sum of prime factors of n (see http://mathworld.wolfram.com/SumofPrimeFactors.html ). There have been observations that certain iterations involving the sopfr function seem to lead invariably to cycles...

According to Bertrand's postulate, there is at least one prime between n and 2n-2, for any n>3. I wonder if there is a theorem about the number of primes between n and 2n exclusive (see http://www.research.att.com/~njas/sequences/A060715 ), because that number seems to be steadily increasing...

I have found something that may be well known, and so I would appreciate either a link or reference, or some help towards proving this: Let f(n) = a*f(n-1) + b*f(n-2), f(0) = 0, f(1) = 1, and g(n) = a*g(n-1) + b*g(n-2), g(0) = A, g(1) = B. Then g(n) = B*f(n) + A*b*f(n-1). The advantage...

u(n) = 6*u(n-1)-u(n-2), u(0)=1, u(1)=1 (this yields the same sequence as with the choices u(0)=1 and u(1)=5) is also the difference sequence of the sequence of the square roots of square triangular numbers: S(n) = 6*S(n-1)-S(n-2), S(0)=0, S(1)=1, that is u(n)=S(n)-S(n-1). This has two...

Thank you. Then, using the generating function method, with a= 3 + 2*sqrt(2), and b = 3 - 2*sqrt(2), A = (-1-sqrt(2)) / (2*sqrt(2)), and B = (1-sqrt(2)) / (2*sqrt(2)), and so u(n) = -A/a^{n+1} - B/b^{n+1}. This works, but it is not as nice as I thought it would be, especially with the...

Thanks for all the help and the useful comments. Sorry for not answering your question earlier, CRGreathouse, but I cannot provide any revealing answer - I think I was playing with the numbers in Excel, searching for patterns. Orthodontist, thanks, but I haven't heard about the method...

I think that the only problem with my proof was saying that q does not divide B(p,q). I would rephrase it this way: Assume p is composite, and q is a prime factor of p. There are q terms in the sequence p, p-1, ..., p-q+1, therefore q divides only one of them, which is p. B(p,q) / p =...