Minimum frequency transmitted by PCB?
Well a printed transformer can (very inefficiently) work at only a few KHz, and there is no issue with placing such things on a board that size, so that implies that your math must be wrong.... 

I could easily build a magnetic loop antenna on that board that would sort of work at 25KHz or so, Q would be rather low and I would worry about dielectric losses and maybe breakdown if running any power into the thing but it would sort of work. 

An odd multiple of a quarter wave is desirable because it makes the feed point appear both resistive and reasonably low impedance, but it is hardly necessary (I can send on 173Khz using a vertical in the garden very much shorter then a quarter wave it is just very inefficient (Think very much less then 1%), and extremely narrowband given the matching network required.