Number-Theory-Inspired Effects in Cold Atoms

Brief overview

The plan is to suggest, using atoms in traps with a tailored spectrum, quantum schemes for a fast verification of several number-theory statements. Among those are the Diophantus-Brahmagupta-Fibonacci identity (that shows that the set of sums of two squares is closed under multiplication) and Goldbach conjecture ("every even is a sum of two primes," yet unproven). Our long-term plan is outlined in [CMC+24]. The project is run in close collaboration with Oleksandr Marchukov of Technische Universität Darmstadt in Germany, Andrea Trombettoni,  Giuseppe Mussardo, and Donatella Cassettari’s experimental group at the University of St. Andrews in Scotland. 

We already successfully tested[MO25]. the sensitivity of one of our protocols on an artificial universe where number 9 was excluded from the set of powers of 3: this hole was instantly detected. The same method will be applied to test if any power of a sum of two squares is a sum of two squares (Diophantus-Brahmagupta-Fibonacci connection) and if a set of sums of two primes contains all the even numbers (Goldbach Conjecture). An alternative Goldbach Conjecture proposal[MTMO24] seeks to achieve a quantum advantage in a search for the violators of the conjecture, using a Grover-inspired protocol.

In a parallel development[C-PPD+24], we assess the role played by the Mersenne numbers in properties of the ground state manifold in the Newman-Moore spin lattices. In another development[RO25], we use Brahmagupta and Perrin identities to first classify and then understand the degeneracies in the spectrum of a 3:1 mass ratio non-interacting particle pair in a box; contrary to the current paradigm, these degeneracies do not seem to be supported by non-commuting symmetries.