Why are there so many primes in the convergents of Pi?

Question

Recently, I was looking into fractional approximations of pi, such as $\frac{22}{7}$ or $\frac{355}{113}$. I found that there was a name for these approximations, 'convergents' of pi, and I found a list of the first 100. Upon inspection of list, it seemed that a lot of the fractions had odd numerators or denominators. Furthermore, it seemed a lot of the fractions had prime numerators or denominators!

A quick check with mathematica confirmed this: 91 out of the first 100 convergents had prime numerators or denominators! Even more astounding, 6 out of the 9 that didn't hold this property occured within the first 11 convergents.

This seems like it can't simply be a coincidence, but I couldn't find anything about it online. Why is this true?

Answer 1

Your data seems to be exactly backwards. Of the first 100 convergents of pi, only 9 of them have either a prime numerator or a prime denominator, with 91 of them having both composite numerator and composite denominator. Moreover, 6 of the 9 that have a prime numerator or denominator are in the first 11 convergents.

Here's the Mathematica code that I used:

In[1]:=   c = Convergents[Pi, 100];

In[2]:=   test = PrimeQ[Numerator[#]] || PrimeQ[Denominator[#]] &;

In[3]:=   Length[Select[c, test]]
Out[3]:=  9

According to Mathematica, these 9 are $$ 3,\;\; \frac{22}{7},\;\; \frac{355}{113},\;\; \frac{103993}{33102},\;\; \frac{833719}{265381},\;\; \frac{4272943}{1360120},\;\; \frac{411557987}{131002976},\;\; \frac{2646693125139304345}{842468587426513207}, $$ $$ \frac{7809723338470423412693394150101387872685594299}{2485912146995414187767820081837036927319426665}. $$