A Padovan prime is a Padovan sequence number 
that is also a prime
number.
The first few distinct Padovan (probable) primes are 2, 3, 5, 7, 37, 151, 3329, 23833, ... (OEIS A100891), and (probable) prime Padovan
numbers occur at indices 
,
4, 5, 7, 8, 14, 19, 30, 37, 84, 128, 469, 666, 1262, 1573, 2003, 2210, 2289, 4163,
5553, 6567, 8561, 11230, 18737, 35834, 44259, 536485, ... (OEIS A112882).
The search for (probable) prime Padovan numbers has been completed up to 
by E. W. Weisstein (Apr. 10, 2011),
and the following table summarizes larger known values.
 | decimal digits | discoverer |
| 469 | 58 | |
| 666 | 82 | |
| 1262 | 154 | |
| 1573 | 192 | |
| 2003 | 245 | |
| 2210 | 270 | |
| 2289 | 280 | |
| 4163 | 509 | |
| 5553 | 679 | |
| 6567 | 802 | |
| 8561 | 1046 | |
| 11230 | 1372 | |
| 18737 | 2289 | |
| 35834 | 4377 | |
| 44259 | 5405 | |
| 536485 | 65518 | E. W. Weisstein (May 16, 2009) |
| 727734 | 88874 | E. W. Weisstein (Apr. 7, 2011) |
