Are there conjectures in math that have been proven wrong although they looked right?

This blog has a series of articles dealing with disproved mathematical conjectures from the history of mathematics. One striking example, is the Mertens conjecture.

Here comes a simpler, but still good example from divisibility and prime number theory – an observation that looks right for quite long, but turns out to be wrong nevertheless:

12 is not a prime number, as you know, because 12=2^2\cdot 3. We now append more and more ones to 12 and check whether this produces a prime number.

• 121 = 11^2 isnot a prime number.
• 1211 = 7\cdot 173 is not a prime number.
• 12111 = 3\cdot 11\cdot 367 is not a prime number.
• 121111 = 281 \cdot 431 is not a prime number.
• 1211111 = 11 \cdot 23 \cdot 478 is not a prime number.
• 12111111 = 3^2 \cdot 31 \cdot 83 \cdot 523 is not a prime number.

Is there a rule here? We check it further. First, however, we introduce a better notation: Thus, we want to denote the number that arises, for example, if we append 10 ones to 12, with 12(1)_{10}. But no matter whether we append 10, 20, 50 or 100 ones, the resulting number is not a prime number:

• 121111111111 = 12(1)_{10}= 61 \cdot 1985428051 – not a prime number
• 12(1)_{20} = 7\cdot 173015873015873015873 – not a prime number
• 12(1)_{50} = 7\cdot 29\cdot 1060502590781\cdot 5625695437821206820171327665134913177 – not a prime number
• 12(1)_{100} = 43487\cdot 24519903637423\cdot 13802420025544751\cdot 8229065925496654663948591534057266582025417047566049794424803802361 – not a prime number

At this point, one might assume that these numbers are never prime numbers. But this is not so. The number which arises, if we append 136 ones to the 12, is then, after all, a prime number! We can have this recalculated, e.g. with Wolfram Alpha:

Well, perhaps this finding is not so surprising to some of us after all. But it does underline the fact that there are patterns which last for an insanely long time before they finally collapse when the numbers are really large. Apart from 12(1)_{136} further prime numbers of this type were discovered, namely

• 12(1)_{184}
• 12(1)_{640}
• 12(1)_{37960}
• 12(1)_{217360}

More info is available online here. The question whether infinite or only finitely many prime numbers of this kind exist is apparently still unsolved.

Euler’s “Lucky Numbers”

The whole matter reminds a little of Euler’s observation (1772) that the polynomial

P(n)=n^2+n+41

for n = 0, 1, 2, \dots , 39 generates 40 primes one after the other. For n=40, however,

\begin{align*}
P(40) &= 40^2 + 40 + 41 \\
          &= 40\cdot 40 + 40 + 41 \\ 
          &= 40\cdot (40+1) + 41 \\ 
          &= 41^2 \\
          &= 1681,
\end{align*}

which is a square number and not a prime number. Here, too, a pattern that has existed for a very long time suddenly loses its validity.