"Every of the three prime residue classes p=1 mod 3 contains an infinite number of primes and the three classes have the densities 1/2, 1/3, and 1/6", (HaH) S. 453

H. Hasse referred to it as "it would be probably more productive for number theory to work on it (cubic Gaussian sums with prime modul p=1mod.3) than working on Fermat's Last Theorem", (HaH), S. 453.

The Kummer conjecture might be addressed by new arithmetic concepts developed in combination with

- the proposed two-semicircle method for binary number theory

- the proposed Krein space based UFT

- the definition of frames for Krein spaces (extending the notion of J-orthogonal bases of Krein spaces), which are the sum of three orthonormal bases of a Krein space, (GiJ), (KaS).

Derbysire's question

All in all the combination of the above thoughs may allow a first answer to the question raised by J. Derbysire, (DeJ) p. 295:

“What on earth does the distribution of prime numbers have to do with the behavior of subatomic particles?“

Related papers

       

Braun, K., The Kummer conjecture and the two-semicircle method

(CoR) Courant R., Hilbert D., Methods of Mathematical Physics, Volume II, Wiley Classics Edition, 1989

(DeJ) Derbysire, Prime Obsession, Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Joseph Henry Press, Washington, D. C., 2003

(GiJ)                

Giribet J. et al., On frames on Krein spaces.pdf

(HaH) Hasse H., Vorlesungen über Zahlentheorie, Springer-Verlag, Belin, Göttingen, Heidelberg, 1950

          

Hasse H., Die Kummersche Vermutung fuer kubische Charaktere

(KaS)              

Karmakar S. et al., Frames on Krein Spaces.pdf