The Strong Law of Small Numbers is an informal statement made by Richard K. Guy: "There aren't enough small numbers to meet the many demands made of them."
More precisely, there are not enough small numbers relative to the multitude of sequences, properties, and patterns that can be defined on the natural numbers. This disparity causes a kind of pigeonholing of sequences over the small numbers, whereby small numbers belong to many different sequences. (To illustrate, search the OEIS for 2, 22, 222, 2222... and compare the number of results for each.) As such, two sequences may coincide for small values before diverging at larger values, and "capricious coincidences cause careless conjectures". In a sense, because their inclusion in an arbitrary sequence is less necessary, larger numbers are more honest witnesses to a sequence's attributes, and "early exceptions eclipse eventual essentials".
|
Statement |
Smallest counterexample |
|---|---|
| Every number is less than a million | Million |
| x2-x+41 has only prime values | x=41 |
| There are always more primes of the form 4k + 3 than of the form 4k + 1 less than n | 26861 |
| Pólya conjecture | 906150257 |
| Every odd-abundant number is divisible by 3 | 5391411025 |
| There are always more primes of the form 3k + 2 than of the form 3k + 1 less than n | 608981813029 |
| n17+9 and (n+1)17+9 are always relatively prime | n=8424432925592889329288197 322308900672459420460792433 |
| For all x π(x)<Li(x) |
1014<x<10316 |