Strong Law of Small Numbers

The first strong law of small numbers (Gardner 1980, Guy 1988, 1990) states "There aren't enough small numbers to meet the many demands made of them."

The second strong law of small numbers (Guy 1990) states that "When two numbers look equal, it ain't necessarily so." Guy (1988) gives 35 examples of this statement, and 40 more in Guy (1990). For example, example 35 notes that the first few values of the interpolating polynomial  (erroneously given in Guy (1990) with a coefficient 24 instead of 23) for , 2, ... are 1, 2, 4, 8, 16, .... Thus, the polynomial appears to give the powers of 2, but then continues 31, 57, 99, ... (OEIS A000127). In fact, this sequence gives the maximal number of regions obtained by joining  points around a circle by chords (circle division by chords).

Similarly, example 41 notes the curious fact that the function  where  is the ceiling function gives the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... (i.e., the first few Fibonacci numbers) for , 1, ..., although it subsequently continues 91, 149, ... (OEIS A005181), which are not Fibonacci numbers.

Another example is provided by a near-identity of trinomial coefficients noticed by Euler.

REFERENCES:

Gardner, M. "Mathematical Games: Patterns in Primes are a Clue to the Strong Law of Small Numbers." Sci. Amer. 243, 18-28, Dec. 1980.

Guy, R. K. "The Strong Law of Small Numbers." Amer. Math. Monthly 95, 697-712, 1988.

Guy, R. K. "The Second Strong Law of Small Numbers." Math. Mag. 63, 3-20, 1990.

Guy, R. K. "Graphs and the Strong Law of Small Numbers." In Graph Theory, Combinatorics, and Applications, Vol. 2 (Kalamazoo, MI, 1988). New York: Wiley, pp. 597-614, 1991.

Sloane, N. J. A. Sequences A000127/M1119 and A005181/M0693 in "The On-Line Encyclopedia of Integer Sequences."