Nontrival primes: human chromosome number is 23,  [Sunflower](http://en.wikipedia.org/wiki/Sunflower) chromosome number is 17, both of which are prime.  Most flowers have an odd and often prime number of petals: five is a common number. 

I assume you really meant **non-trivial primes**, where $p>2$ or $p>3$, but you didn't specify that, so let me point out some basic regions where 2 and 3 are prominent.  And $2$ and $3$ are very prominent through-out nature, so these are not very special concepts.

Diploid genomes come in copies of 2, and 2 is prime.  DNA chains come in duplicate copies, with one side reading in one direction as the **sense** and the other side being the "inverted carbon copy" and called the **nonsense** side.  When DNA chains are copied, they split apart like a zipper and complementary copies are made on both, yielding two identical doubled-DNA chains.

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Three is also a prime number:

The human chromosomal set consists of 23 pairs of DNA chromosomes: $46$ in total, with $44$ of those being non-sex chromosomes, and $2$ of them being sex chromosomes: $XX$ for females and $XY$ for males.


Also, when you convert from 2 copies of these chromosomes to three, for example [Trisomy 21](http://en.wikipedia.org/wiki/Trisomy_21), you can end up with Down's syndrome.

If, instead of having two sex chromosomes of the usual type, $XX$ or $XY$, you can have [XYY syndrome](http://en.wikipedia.org/wiki/XYY_syndrome) or [XXY syndrome also known as Klinefelter's syndrome](http://en.wikipedia.org/wiki/Klinefelter%27s_syndrome).  Technically, you could say that these types of sex-chromosome sets are not in the set of the usual two genders of male ($XY$) and female ($XX$), but are actually outside of the two genders.  So to answer @vonjd's comment to the question, there are not just 2 genders. 

Extra random point: the sunflower's spiral follows [Fermat's spiral](http://en.wikipedia.org/wiki/Fermat%27s_spiral) which at various points, is a prime number.  So there must be some sunflowers which have a prime number of sunflower seeds.

Many flowers have five petals, five is prime.