Most functions do not have any fixed points, and many have only one such point. (Recall that x is a fixed point of a function f if f(x) = x.)
The situation appears to be much rosier for "fixated points" of arithmetical functions. For simplicity, an arithmetical function in this discussion refers to a function having domain and range which are sets of integers. n is a fixated point of an arithmetical function f if, treated as strings, n and its reversal n' are both substrings of f(n). (The reason for the name "fixated" is that n appears fixated with itself. :-))
The notion of fixated points can be seen as a generalization of palindronomic numbers (that is, numbers which remain the same when reversed). This is because the palindronomic numbers are exactly the fixated points of the identity function id(n) = n which do not end in the digit 0 (Those that end in 0 lose the 0 when reversed since leading 0's are ignored.)
For example, consider f(n) = 2n. Then n = 49 is a fixated point of f since f(49) = 249 = 562949953421312 in which both 49 and its reversal 94 appear as substrings. The sequence of fixated points of f begins
6, 10, 44, 49, 60, 67, 190, 191, 226, 252, 321, 373, ....
To take another example, let f(n) = Fibonacci(n), the n-th Fibonacci number. (Recall that f is defined recursively by f(1) = f(2) =1, f(n) = f(n-1) + f(n-2) for n > 2.) Then n = 43 is a fixated point of f since f(43) = Fibonacci(43) = 433494437 in which both 43 and its reversal 34 appear as substrings. The sequence of fixated points of f begins
1, 5, 41, 43, 65, 70, 85, 99, 101, 194, 214, 340, 368, ....
The reader can verify as an exercise that the sequence of fixated points of f(n) = n3 has first few terms
1, 4, 5, 6, 9, 10, 40, 50, 60, 90, 99, 100, 400, 500, ....
and the sequence of fixated points of f(n) = n4 begins
1, 5, 6, 10, 50, 60, 92, 100, 363, 500, 600, 636, 909, 1000, ....
As a rule of thumb, the faster f(n) grows with respect to n, the more digits f(n) will have as compared with n, hence, the likelier it is to contain n and n' as substrings. Thus, it is more interesting to consider f(n) which grows slowly with respect to n. One such function is given by f(n) = Prime(n), the n-th prime, which has a rate of growth with respect to n that is approximately equal to ln(n) (the natural logarithm of n). The only fixated points of f not exceeding 107 are 7 and 6460. Prime(7) = 17, which contains both 7 and its reversal 7; and Prime(6460) = 64601, which contains both 6460 and its reversal 646 (= 0646). Can the reader find any more such points?
To aid in the search, here is Mathematica code to generate fixated points:
r = {};
Do[m = Prime[n];
s = ToString[m];
If[StringPosition[s, ToString[n]] != {} &&
StringPosition[s, ToString[FromDigits[Reverse[IntegerDigits[n]]]]] != {},
r = Append[r, n]], {n, 1, 10^10}];
rTo use this code for a different function, just replace the "m = Prime[n]" by the appropriate expression.
Joseph L. Pe
iDEN System Engineering Tools and Statistics
Motorola
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Deer Park, IL 60010
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©2002 J. L. Pe. Document created on 27 March 2002 by J. L. Pe. Last updated on 1 May 2002.