09-02-2026, 11:07 PM
Summary
The Strong Law of Small Numbers, introduced by mathematician Richard K. Guy, is a humorous but useful warning about mathematical patterns: there simply are not enough small integers to avoid them appearing repeatedly in seemingly unrelated situations. As Guy put it, “There aren't enough small numbers to meet the many demands made of them.” Consequently, striking coincidences involving numbers such as $1,2,3,4,\dots$ may look profound even when they are simply consequences of the limited supply of small numbers. Guy developed this idea in his influential 1988 paper The Strong Law of Small Numbers.
Guy later proposed a Second Strong Law of Small Numbers: when two numerical patterns appear to agree, they may eventually diverge. Observing only the first few terms of a sequence can therefore lead to false conjectures. One example involves numbers of the form $2^p-1$: for the first prime values $p=2,3,5,7$, the results are prime, which might suggest that $2^p-1$ is always prime whenever $p$ is prime. But the pattern already fails at $p=11$, since $2^{11}-1=2047=23\times89$. Another famous example is Moser's circle problem, where the maximum number of regions created by joining $n$ points on a circle follows $1,2,4,8,16$ for the first five cases, tempting one to guess $2^{n-1}$; at $n=6$, however, the correct value is $31$, not $32$.
The broader lesson is methodological: small numerical examples are excellent for discovering conjectures but weak evidence for proving them. Mathematical patterns can survive surprisingly many initial cases before breaking down, so genuine proof is essential before treating an observed pattern as a general law.
Key takeaways
ARTICLE
The Strong Law of Small Numbers, introduced by mathematician Richard K. Guy, is a humorous but useful warning about mathematical patterns: there simply are not enough small integers to avoid them appearing repeatedly in seemingly unrelated situations. As Guy put it, “There aren't enough small numbers to meet the many demands made of them.” Consequently, striking coincidences involving numbers such as $1,2,3,4,\dots$ may look profound even when they are simply consequences of the limited supply of small numbers. Guy developed this idea in his influential 1988 paper The Strong Law of Small Numbers.
Guy later proposed a Second Strong Law of Small Numbers: when two numerical patterns appear to agree, they may eventually diverge. Observing only the first few terms of a sequence can therefore lead to false conjectures. One example involves numbers of the form $2^p-1$: for the first prime values $p=2,3,5,7$, the results are prime, which might suggest that $2^p-1$ is always prime whenever $p$ is prime. But the pattern already fails at $p=11$, since $2^{11}-1=2047=23\times89$. Another famous example is Moser's circle problem, where the maximum number of regions created by joining $n$ points on a circle follows $1,2,4,8,16$ for the first five cases, tempting one to guess $2^{n-1}$; at $n=6$, however, the correct value is $31$, not $32$.
The broader lesson is methodological: small numerical examples are excellent for discovering conjectures but weak evidence for proving them. Mathematical patterns can survive surprisingly many initial cases before breaking down, so genuine proof is essential before treating an observed pattern as a general law.
Key takeaways
- The Strong Law of Small Numbers is a mathematical observation and joke, not a formal theorem.
- Small integers recur so frequently that apparently remarkable coincidences should be treated cautiously.
- A formula fitting the first several cases of a sequence may eventually fail.
- Examples suggest conjectures; proofs establish them.
ARTICLE
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