# Examples of patterns that eventually fail

Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of "proof". I receive responses like: "surely if Collatz is true up to $20×2^{58}$, then it must always be true?"; and "the sequence of number of edges on a complete graph starts $0,1,3,6,10$, so the next term must be 15 etc."

Granted, this second statement is less logically unsound than the first since it's not difficult to see the reason why the sequence must continue as such; nevertheless, the statement was made on a premise that boils down to "interesting patterns must always continue".

I try to counter this logic by creating a ridiculous argument like "the numbers $1,2,3,4,5$ are less than $100$, so surely all numbers are", but this usually fails to be convincing.

So, are there any examples of non-trivial patterns that appear to be true for a large number of small cases, but then fail for some larger case? A good answer to this question should:

1. be one which could be explained to the layman without having to subject them to a 24 lecture course of background material, and
2. have as a minimal counterexample a case which cannot (feasibly) be checked without the use of a computer.

I believe conditions 1. and 2. make my question specific enough to have in some sense a "right" (or at least a "not wrong") answer; but I'd be happy to clarify if this is not the case. I suppose I'm expecting an answer to come from number theory, but can see that areas like graph theory, combinatorics more generally and set theory could potentially offer suitable answers.

• The sentence: ""the numbers 1,2,3,4,5 are less than 100, so surely all numbers are" - Is interesting. – NoChance Feb 20 '12 at 22:01
• @yasmar, I was thinking of this: mathoverflow.net/questions/15444/… – Gerry Myerson Feb 20 '12 at 22:26
• This doesn't satisfy b), but how about "$n^2-n+41$ is always prime"? (it's true for $1\le n\le 40$). – David Mitra Feb 20 '12 at 22:39
• @EmmadKareem After reading halfway through this page, this looks like a challenge to see who can give the most mind blowing example of this simplified version: "N not equals 82174583229565384923 for N = 1,2,3,4... breaks down at N = 82174583229565384923" – Jake Feb 22 '12 at 13:27
• $e = 2.7 \, 1828 \, 1828 \quad$ :O $\quad 459045235 \,$ :( – Lenar Hoyt Jul 14 '13 at 22:48

The Sierpiński numbers would be a good example. All odd integers up to 10,221 have been checked and are known to lead to a prime number of the form $k2^n+1$, where $k$ is the original odd integer, and $n$ is any integer. One would think that, if the trend continued, there would be no such integers. However, several integers have been proven to generate only composite numbers of the form $k2^n+1$. The smallest such integer known is $78,\!557$. In addition, it has been proven that there are infinitely many such integers.

You have not died every day since you were born.

• Well, that's kind of true and maybe kind of funny, if you like dark humor. The OP was most likely more interested in something technical, so I fear your memento mori is not answer (let alone the fact that it's off-topic). – user228113 May 25 '15 at 23:05
• My humor is 71% dark. It may not be the best tasting, but it is better for you. – marty cohen May 25 '15 at 23:52
• Well, it's related to logic. Not very technical, but easy to understand. – GregT Apr 18 '18 at 9:34

Another example, hopefully not too technical for a layman, is the story of Skewes' number (see link) where there was, again, a lot of numerical evidence that $\pi(x)$ was always less than $\operatorname{li}(x)$ - until Littlewood proved that this was not the case and Skewes was the first one to establish bounds for the smallest counterexample.

• Quoting James Grime on Numberphile: "So this is a pattern that holds ... into huge sizes of googols of googols of googols, and it appears that this inequality holds and then it flips." ( youtu.be/Lihh_lMmcDw?t=549 ) – Brian Jan 30 at 0:52

Is there a pair of amicable numbers with distinct smallest prime factors? For quite a long time no example of such a pair was known. The first such pair was discovered through computer search in October 2015 (but only noticed by humans three months later):$$445\,953\,248\,528\,881\,275=3^2\times5^2\times7\times13\times19\times37\times43\times73\times439\times22\,483$$and$$659\,008\,669\,204\,392\,325=5^2\times7\times13\times19\times37\times73\times571\times1\,693\times5\,839.$$

One good example of such a sequence is a prime number generating sequence related to the Ulam spiral: $N^2 - N + 41$. Tt generates prime numbers when N = any integer from 0 to 40 inclusive, but obviously fails at 41, returning the result of 41^2. The solutions form a diagonal line on an Ulam spiral that starts at 41

Here is a Youtube video by Numberphile on the sequence:

Let $$p_n$$ be the n$$^{th}$$ prime. Consider n$$^{th}$$ prime raised to $$1/n^{th}$$ power, i.e. $$p_n^{\frac{1}{n}}$$. Firoozbakht conjectured that $$p_n^{\frac{1}{n}}$$ is a strictly decreasing function. But, mathematicians believe it to be false. This may become one the best examples of such patterns if and when it is proven to be false.