Prime numbers -- your gems

Prime numbers or primes are man's eternal treasures that have been cherished for several millennia, until today. As their academic ancestors in ancient Mesopotamia, many mathematicians are still trying hard to see primes better. I shall relate here a part of my impressions that I have gathered in a corner of my mind through my own research and excursions outside my profession. This is in essence a translation of my Japanese article that was prepared for my public talk at the general assembly of the Mathematical Society of Japan, Spring 2005, and is now available on the home page of MSJ. At this opportunity I have made substantial revisions, and changed the title into a more appropriate one. This is also an enlarged version of my public talk to be delivered at TECHNION, Haifa, on Dec 20, 2005.