Quantum abacus generates prime numbers

Physicists develop the first quantum system for generating arithmetic series of numbers

Prime numbers as quantum states: For the first time, physicists have constructed a quantum system that can determine and represent prime numbers and other mathematical series of numbers. Based on special formulas, this quantum “abacus” generates an optical interference pattern in which, for example, only the prime numbers appear as discrete energy states. According to the team, such quantum systems could open up new possibilities for calculating arithmetic series of numbers.

Prime numbers are one of the most fascinating phenomena in mathematics. Because these numbers are only divisible by one and themselves – and thus the “atoms” of the world of numbers. At the same time, they appear to be randomly and unpredictably distributed in the string of numbers, and their sequence cannot be calculated using a simple formula. This is one of the reasons why prime numbers form the basis of most encryption . In addition, mathematicians are still trying to unravel the subtle patterns in the infinite series of prime numbers and to find ever larger prime numbers .

Quantum Abacus
Structure of the quantum abacus. The specific pattern of the photonic liquid crystal lattice creates optical quantum dots in a light trap whose energy levels correspond to the prime numbers.© Cassettari et al./ PNAS Nexus, CC-by 4.0

“Abacus” for prime numbers

Now physicists have developed a whole new method for generating prime numbers – a quantum abacus. The team led by Donatella Cassettari from the University of St. Andrews in Great Britain wanted to know whether the seemingly random sequence of prime numbers and other arithmetic series of numbers could be determined and represented using quantum systems. The goal was to develop a quantum system that represents the prime numbers in the form of discrete energy levels.

“Given the important role prime numbers play in many mathematical problems, from factoring integers to famous conjectures such as the Riemann hypothesis or the Goldbach conjecture, it would be advantageous if prime numbers were also investigated in new ways through the experimental control of quantum systems could be generated,” state the researchers. The prime numbers or other series of numbers ultimately arise as a result of physical laws.

Energy levels of optical quantum dots as indicators

In principle, this system is a quantum abacus, as the team explains. Because similar to how an abacus enables calculations using simple mechanical movements of balls, a quantum abacus uses quantum mechanical manipulations for the calculation. In concrete terms, the system consists of a holographic structure in which a liquid crystal grating modifies an incident laser beam in a specific way.


The shape of the liquid crystal lattice and the additional lenses and mirrors are chosen in such a way that an interference pattern with a sequence of quantum dots is created in this light trap. These quantum dots have energy states whose intensity values ​​result in a sequence of numbers – in one experiment these were the first prime numbers, in another the “lucky numbers”, another similarly unpredictable sequence of numbers.

Also suitable for other arithmetic series of numbers

“We have already calculated and generated the quantum potentials of the first 15 prime numbers in our laboratory, as well as those of the first ten lucky numbers,” reports co-author Giuseppe Mussardo from the SISSA research center in Trieste. “But the same technique can also be applied to other sequences of numbers – including infinitely long ones.” The only restriction is that the numbers in the sequence must not increase too quickly. “The growth of the infinite series must be less than N squared,” explains Mussardo.


According to the researchers, their quantum abacus is a first step towards novel quantum systems with which algorithms important for quantum technologies can be tested and developed in new ways. “Our work shows the feasibility of this approach and paves the way for exploring mathematical problems and arithmetic manipulations using quantum mechanical experiments,” says co-author Andrea Trombettoni from the University of Trieste. (PNAS Nexus, 2023; doi:10.1093/pnasnexus/pgac279 )

Source: Scuola Internazionale Superiore di Studi Avanzati

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