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We construct an analogue computer based on light interference to encode the hyperbolic function f (ζ) ≡ 1/ζ into a sequence of skewed curlicue functions. The resulting interferogram when scaled appropriately allows us to find the prime number decompositions of integers. We implement this idea exploiting polychromatic optical interference in a multi-path interferometer and factor seven digit numbers. We give an estimate for the largest number that can be factored by this scheme.To find the factors of a large integer number N is a problem of exponential complexity. Indeed, the security of codes relies on this fact but is endangered by Shor's algorithm [1], which employs entanglement between quantum systems [2]. In the present paper we report the optical realization of a new algorithm for factoring numbers, which takes advantage of interference only.A naive way of approaching the problem of factorization consists of dividing N by integers ℓ, starting from ℓ = 3 until N/ℓ is an integer. In the worst case this procedure requires √ N divisions before one would find a factor. On a digital computer division of large numbers is a rather costly process. However, in many physical phenomena division occurs in a rather natural way. For example, a wave of wavelength λ, propagating over a distance L, acquires a phase φ = 2πL/λ and therefore probes the ratio L/λ. In the optical domain λ is measured in nanometers (nm), that is λ = ℓ nm. When we also express the path length L in units of nm, that is L = N nm, the phase φ = 2πN/ℓ is sensitive to the ratio N/ℓ. For factors of N , φ is an integer multiple of 2π. Otherwise φ is a rational multiple of 2π.In order to enhance the signal associated with a factor relative to the ones corresponding to non-factors, we use interference of waves, which differ in their optical path length by an integer multiple. In this way we take advantage of constructive interference when ℓ is a factor of N , but destructive interference when ℓ is not a factor of N . The cancellation of terms is most effective when the individual optical paths increase in a nonlinear way. In this case, the intensity of the interfering waves is determined by the absolute value squared of a truncated exponential sum [3]. A polychromatic source of light, which contains several wavelengths λ = ℓ nm, allows us to test several trial factors simultaneously, taking advantage of the properties of truncated exponential sums [4] with continuous arguments.Our method is motivated by recent work on factorization using truncated exponential sums [5], which has been realized in several experiments [6]. However, it differs from the past realizations in three important points:(i) the division of N by the trial factors ℓ is not precalculated [7], but it is performed by the experiment it- self, (ii) all the trial factors are tested simultaneously in a single experiment, and (iii) a scaling property inherent in the recorded interferogram of a single number allows us to obtain the factors of several numbers.The optical setup used to implement th...

In this paper, we will describe a new factorization algorithm based on the continuous representation of Gauss sums, generalizable to orders j > 2. Such an algorithm allows one, for the first time, to find all the factors of a number N in a single run without precalculating the ratio N/l, where l are all the possible trial factors. Continuous truncated exponential sums turn out to be a powerful tool for distinguishing factors from non-factors (we also suggest, with regard to this topic, to read an interesting paper by S. Wölk et al. also published in this issue [Wölk, Feiler, Schleich, J. Mod. Opt. in press]) and factorizing different numbers at the same time. We will also describe two possible M-path optical interferometers, which can be used to experimentally realize this algorithm: a liquid crystal grating and a generalized symmetric Michelson interferometer

We will describe a new factorization algorithm based on the reproduction of continuous exponential sums, using the interference pattern produced by polychromatic light interacting with an interferometer with variable optical paths. We will describe two possible interferometers: a generalized symmetric Michelson interferometer and a liquid crystal grating. Such an algorithm allows, for the first time, to find all the factors of a number N in a single run without precalculating the ratio N/l, where l are all the possible trial factors. It also allows to solve the problem of ghost factors and to factorize different numbers using the same output interference pattern.

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