The Strong Law of Small Numbers was the title of a paper written by R.K. Guy, in which he asserts that "there are not enough small numbers to satisfy all the demands placed on them." In other words, many things are not true even though they are true for every number that you try, because the first counterexample has 20 digits, or maybe 10²⁰ digits. We can come up with so many different "properties" for numbers to have that we come up with apparent patterns that hold for all the numbers we can conveniently do calculations with, but simply aren't true of all numbers. One example:

gcd(n¹⁷+9, (n+1)¹⁷+9) seems to always be one. In fact, if you had your computer checking this for n=1, 2, 3, . . . successively, it would never find a counter-example. That is because the first counter-example is

8424432925592889329288197322308900672459420460792433

Stolen from http://primes.utm.edu/glossary/page.php/LawOfSmall.html

This law explains why it's always necessary to prove things, rather than assuming them to be true (like that old urban legend about pi being normal.)

Reference:
R.K. Guy. "The Strong Law of Small Numbers." Amer. Math. Monthly 95 (1988): 697-712