833719/265381

Based on the simple continued fraction of π, its convergents (rational approximations) are: 3/1, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, 208341/66317, 312689/99532, 833719/265381, 1146408/364913, ...

Prime numerators are at position 1, 5, 9, ... Prime denominators are at position 2, 4, 9, ... The ninth convergent therefore has both prime numerator and prime denominator, noted ~2003 in the OEIS. It seems unlikely that we will ever see another such.

I thought it might be useful to have here a listing of the positions of prime numerators (p/) and prime denominators (/p) so as to better assess the rarity of their confluence:

1/
/2
/4
5/
9/
/9
11/
16/
/33
87/
230/
334/
594/
/595
840/
853/
/1127
1149/
/2003
2726/
/3611
3788/
/4356
/6926
7442/
8751/
/25333
/27652
/32395
/37722
42038/
/114199
143753/
...