559 = 13 * 43
If something is 344 (mod 559), it is the same as simultaneously being 6 (mod 13) and 0 (mod 43) ( read up on Chinese Reminder Theorem to understand this equivalence)
Periodicity of 34^x (mod 13) is 4, and (mod 43) is 42.
Wait! 4 doesn't divide 42. Doesn't look like your observation is true.
Try 23*34^44 +5590*y+588. This is 0 (mod 559). Disproved!
