Strong law of small numbers

file:circle_division_by_chords.svg as an example. The number of {{nowrap|points (n),}} {{nowrap|chords (c)}} and {{nowrap|regions (r_G)}}. The first five terms for the number of regions follow a simple sequence, broken by the sixth term.]]

Guy also formulated a second strong law of small numbers:

{{quote|When two numbers look equal, it ain't necessarily so!{{cite journal

| last = Guy | first = Richard K. | author-link = Richard K. Guy

| doi = 10.2307/2691503

| issue = 1

| journal = Mathematics Magazine

| jstor = 2691503

| pages = 3–20

| title = The second strong law of small numbers

| volume = 63

| year = 1990}}}}

Guy explains this latter law by the way of examples: he cites numerous sequences for which observing the first few members may lead to a wrong guess about the generating formula or law for the sequence. Many of the examples are the observations of other mathematicians.

One example Guy gives is the conjecture that $2^p-1$ is prime—in fact, a Mersenne prime—when $p$ is prime; but this conjecture, while true for $p$ = 2, 3, 5 and 7, fails for $p$ = 11 (and for many other values).

Another relates to the prime number race: primes congruent to 3 modulo 4 appear to be more numerous than those congruent to 1; however this is false, and first ceases being true at 26861.

A geometric example concerns Moser's circle problem (pictured), which appears to have the solution of $2^\{n-1\}$ for $n$ points, but this pattern breaks at and above $n=6$.

• Insensitivity to sample size
• Law of large numbers (unrelated, but the origin of the name)
• Mathematical coincidence
• Pigeonhole principle
• Representativeness heuristic

{{Reflist}}

• {{Cite web

|first=Chris

|last=Caldwell

|title=Law of small numbers

|url=http://primes.utm.edu/glossary/page.php?sort=LawOfSmall

|work=The Prime Glossary

}}

• {{MathWorld|urlname=StrongLawofSmallNumbers|title=Strong Law of Small Numbers}}
• {{Cite web

|title=Small finite sets

|work=Secret Blogging Seminar

|date=2007-10-27

|first=Scott

|last=Carnahan

|url=http://sbseminar.wordpress.com/2007/10/27/small-finite-sets/

|postscript=, notes on a talk by Jean-Pierre Serre on properties of small finite sets.

}}

• {{cite journal |title=Belief in the law of small numbers. |author1=Amos Tversky |author2=Daniel Kahneman |journal=Psychological Bulletin |volume=76 |number=2 |pages=105–110|date=August 1971 |doi=10.1037/h0031322 |quote=people have erroneous intuitions about the laws of chance. In particular, they regard a sample randomly drawn from a population as highly representative, I.e., similar to the population in all essential characteristics.|citeseerx=10.1.1.592.3838 }}

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