1234567891, 12345678901234567891, and 1234567891234567891234567891 are prime.

So are

19
197
1979
19793
197933
1979339
19793393 and
197933933.

And so are

742950290870000078092059247
742950290871010178092059247
742950290872020278092059247
742950290873030378092059247
742950290874040478092059247
742950290875050578092059247
742950290876060678092059247
742950290877070778092059247
742950290878080878092059247 and
742950290879090978092059247.

If the nth term of the Fibonacci series is prime, then n also is prime (where n > 4). For example, the 17th term, 1597, is prime, and 17 is prime.

05/27/2017 UPDATE: Further to the first series, 1979339333 is also prime! (Thanks, Alon.)