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    We demonstrate a new method for prime number decomposition using the Talbot effect, a phenomenon from optics. This approach leverages optical properties to factorize composite numbers, offering a novel technique in number theory.

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    Area of Science:

    • Optics
    • Number Theory
    • Computational Mathematics

    Background:

    • The Talbot effect is a classical optics phenomenon related to wave propagation.
    • Gauss sums are a mathematical tool in number theory, useful for analyzing prime numbers and factorization.
    • A connection exists between the Talbot effect and Gauss sums, suggesting potential for optical number theory applications.

    Purpose of the Study:

    • To implement a novel algorithm for prime number decomposition.
    • To utilize the Talbot effect and its relationship with Gauss sums for factorization.
    • To experimentally verify a new optical approach to prime decomposition.

    Main Methods:

    • Employing the Talbot effect, a near-field optics phenomenon.
    • Utilizing the mathematical framework of Gauss sums.
    • Analyzing the longitudinal intensity profile of the Talbot carpet for prime decomposition.
    • Experimental verification of the proposed algorithm.

    Main Results:

    • Successful demonstration of prime number decomposition using the Talbot effect.
    • Validation of a novel algorithm based on optical intensity profiles.
    • Identification and discussion of the limitations of the developed approach.

    Conclusions:

    • The Talbot effect provides a viable method for prime number decomposition.
    • The connection between optics and number theory can be exploited for computational tasks.
    • The developed algorithm offers a new perspective on number theoretic problems.