TITLE: What are conjectures that are true for primes but then turned out to be false for some composite number?
QUESTION [10 upvotes]: Note: This is an update formulation since many people misunderstood the question before.
Of course it is easy to make a statement like "Every n is a prime or at most 1000", which is true for every prime $n$ and every small $n$ but fails for $n=1002$. What are "real" conjectures that were known to hold for primes and small values, then turned out to be false?
An excellent example about cyclotomic polynomials was given by Aaron in the comments. Here the conjecture was that the coefficients are $0, \pm 1$ for every $n$. This holds for primes and small $n$'s, but fails for $105$.
Also the existence of Carmichael numbers comes close, but here the problem itself involves primes, I would like something less "primey". I know conjuctures that are or were known only for primes. Recently solved is Colorful Tverberg, still unknown is Evasiveness or this little MO problem.

REPLY [6 votes]: An aliquot sequence is a recursive sequence in which each term is the sum of the proper divisors of the previous term. Catalan conjectured that no aliquot sequence is unbounded. The conjecture is trivially true for primes, since the aliquot sequence for $p$ is $p,1,0$ (terminating, since the sum of the proper divisors of zero is undefined). The conjecture is also true, by computation, for all $n\le275$. Strictly speaking, this doesn't qualify as an answer to the question, since there is no $n$ for which the conjecture has been proved to be false, but there are so many integers for which the aliquot sequence shows no sign of ever reaching a bound that it seems most plausible to conclude that the conjecture is false. 
Here is a website dedicated to the problem.

REPLY [4 votes]: An important question in matroid structure theory is given a matroid $M$, how many inequivalent representations does it have over a fixed finite field $\mathbb{F}$?  Two $\mathbb{F}$-matrices $A$ and $B$ are equivalent if they are related by a projective transformation, in which case they clearly represent the same matroid.  
In 1988, Kahn proved that $3$-connected $GF(4)$-representable matroids have a unique representation over $GF(4)$.  In the same paper, he conjectured that for every prime power $q$, there is an $n(q)$, such that every $3$-connected $GF(q)$-representable matroid has at most $n(q)$ inequivalent representations. Oxley, Vertigan and Whittle proved Kahn's conjecture for $GF(5)$, but also showed that it fails for all larger finite fields.  This naturally led to the question of what happens for $4$-connectivity, and the following weakening of Kahn's Conjecture. 
Conjecture 1. For every prime power $q$, there is an $n(q)$ such that every $4$-connected $GF(q)$-representable matroid has at most $n(q)$ inequivalent representations over $GF(q)$.
One would expect that Conjecture $1$ holds for all finite fields, or fails for sufficiently large finite fields. However, remarkably there is a dichotomy depending on whether $q$ is prime!  
Theorem 2. (Geelen and Whittle) For every prime $p$, there is an $n(p)$ such that every $4$-connected $GF(p)$-representable matroid has at most $n(p)$ inequivalent representations over $GF(p)$.
The proof of Theorem $2$ is not easy; the paper is $175$ pages.  
Theorem 3. (Geelen, Gerards and Whittle)
For every non-prime power $q \geq 9$, there is a $4$-connected $GF(q)$-representable matroid with arbitrarily many inequivalent representations over $GF(q)$.