Prime divisors of polynomials, spin chains, and non-residues

The project aims to contribute to unearthing various deep mysteries of arithmetic. For instance, the infinitude of the sequence 2, 3, 5, 7, 11, 13, 17, of prime numbers has been known already to Euclid more than 2250 years ago, yet even innocuous-sounding questions regarding the existence of infinitely many prime numbers of certain special forms keep persisting to stubbornly resist any attempt at their resolution to this very day. (A prime number is a positive integer larger than 1, whose only positive divisors are 1 and the number itself.) For example, it is still not known whether the sequence 2, 5, 17, 37, 101, 197, 257, of prime numbers, that are larger than a square by exactly one, progresses infinitely or whether it ceases eventually. In 1912, Edmund Landau, an eminent mathematician of the 20th century, called this and three other problems unassailable at the present state of [mathematics]. Despite the fact that the intractability of the original problems does not seem to have withered in the slightest over the past century, a number of ways have been paved by whose future extensions one hopes to one day surmount the intractable. The present project aims to pave further terrain along those roads. Alongside primes, number theorists share a keen interest into various other types of numbers and their distribution. Of particular interest are, for instance, questions pertaining to the arithmetic on analogue watches and its extension to successively finer watch faces. (As an example, one studies questions related to the arithmetic in which 5+8=1, in the same way in which 5 hours past 8 o`clock, the hour hand on a watch shows 1.) Conversely, wrapping the integers around a watch face, often allows one to gain important insights into the nature of the integers themselves. A classical question is how square numbers on watches with densely numbered watch faces distribute. (Here squareness is to be understood with respect to the watch arithmetic.) While such squares are certainly not the only concern, questions about them should be understood as prototypes for many other problems, and progress on the former is often seen as a measure for the present state of knowledge on the whole. Also with regard to distribution of such squares there is a wide gap between what is generally conjectured to be true, and what one has hitherto been able to prove. In particular, there is a large discrepancy in what one knows about the distribution of watch squares about midnight rather than about a arbitrary points. The present project seeks to develop new methods capable of surmounting the intrinsic limitations of earlier methods in an effort to reduce the aforementioned discrepancy.