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that by a physical realization into natural numbers (the set N) using the Euler-Lagrange equation [20].

Only physical frequencies (in the set N) do exist. We see that visually, in a continued fraction expansion, in a harmonic composition using the DFT [20], or as well-known in music.

These physical frequencies are seen, also experimentally, following an extended sum partition (as a linear combination of independent oscillators [20]), but not as a product.

In the case of 6, one can see oscillations in 3 sets of 2, or 2 sets of 3. There is no physical space for any other combination.

2. NON-BOOLEAN LOGIC

QC has been instrumental in the realization of Section 1 and its sub-sections, using non-Boolean logic and allowing us to consider the sets R=C=Q in numerical calculations, but

Boolean logic can be used to an even greater effect and faster calculations, with no confounding factors. This will be published elsewhere, deprecating tri-state™ as necessary in hardware, as considered by Intel and others.

QC uses simultaneous multiple-states logic (following ‘all states at once’), not only Boolean logic. To those who question that

QC, offering many more states of logic would be somehow “illogical” to consider, one notes that, in unpublished notes, before 1910, Charles Sanders Pierce is well-known to have soundly rejected the idea that all propositions must be either

True or False, as in Boolean logic, the same as Frege is well-known to have proposed in mathematical semantics.

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Pierce developed well-understood rules where the LEM2 is not valid, including some truth tables.

The three logical states are: 1 (true), 0 (false), and X (undefined), as well-known. This avoids the Gödel uncertainties

(to be published elsewhere). In QC, one can be more precise and include more states than physical QM (e.g., quantum annealing) as one makes the model as the behavior be more inclusive, viewed internally, even though externally one should be limited to use the LEM3. Hence, QC using software promises to be easier to delve deeper (e.g., one googol) than

QM using hardware in the number of states, while it can use available hardware -- as we report (e.g., cellphone, desktop PC).

This is an important result for the digital industry, which uses Boolean logic in hardware, and for the future of QC.

Our empirical conjecture #4 is that: “We can achieve freedom from the LEM in the behavior while the implementation can obey the LEM.”

2 This can be further understood, by noting that the LEM says [22] that “For every proposition ‘p’, either ‘p’ or ‘not p’ holds”. To many this is a “self-evident” truth, but only works in a binary logical system, such as Boolean logic. For example, it does not apply to ’p’ and ’not p’, a valid case in logic, but not in binary logic.

3 Other researchers may at first diverge from this interpretation. The LEM has been an inescapable bedrock of logical reasoning in binary logic. But any digital circuit supporting 3 or more logical states, naturally breaks the LEM. Thus, LEM or even tri-state™ can always be obeyed externally, by a physical constraint on the digital circuits, such as a FPGA or a 74LS241 octal buffer. Albeit, it can be designed with multiple states ( e.g., tri-state™, ‘all states at once’ in QC) in mind. A medical doctor can use QC in his mind while treating classically for a pathology, using available means.

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Thus, tri-state™ can be abandoned by Intel and others, favoring Field Programmable Gate Arrays (FPGA) [cf. 21].

A software breaking the LEM can run on LEM hardware [22].

The (multiple) existing states are in different dimensions, and a continuous path in a higher dimension (e.g., isolated three states in tri-state™, simultaneous multiple-states in QC) must necessarily map into a discontinuous path in a lower dimension (digital hardware, binary).This happens due to a well-known theorem in topology and projection, that [22] calls Topological

Relationship (TR). See footnote 2.

3. THE OBJECTIVE WAVE MODEL

The DFTM proposes an objective wave model for vibrations, as in subsection 1.4, linking objective periodicity with the factoring of integers, to find prime numbers -- using, e.g., the DFT.

Here, interference makes it easier to measure periodicity objectively, and this should not come as a big surprise. See footnote 5.

Physicists routinely use scattering of physical waves and interference measurements to determine periodicity of physical objects such as crystal lattices.

In an important well-known work, Bennett (the physicist Charles

Bennett, 1973) showed that any classical computation can be transformed into a reversible form, which allows quantum mechanics (QM) to be used in computation, creating quantum computation QC!

This answers the first question of this work, see Introduction:

(1) what role can QM have in QC? The answer is: QM can be

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used in QC. We go from physics to mathematics, as in a “wormhole”! Thanks mainly to Charles Benett.

With the objective wave model, we provide the tools (and some are used here) to answer the second question of this work, see

Introduction: (2) what may QC reveal?

We can use localized, objective oscillations in a DFTM, quantifying both the behavior of chords in terms of physical waves (and frequencies) and of objective numbers -- in a 1:1 model. See footnote 5.

4. THE GCD

The GCD plays a fundamental role in our QC technique. The GCD allows one to use an efficient method for computing the greatest common divisor (GCD) of two integers, to be used here. The GCD is the largest number that divides them both exactly. But it operates with each digit at a time -- although it can be parallelized (see Section 8), it remains non quantum.

Prime numbers can be found by a well-known method, with slow convergence, dating back to Euclid; the Euclidean algorithm -- a way to find the GCD -- named after the ancient Greek mathematician Euclid, who first described it in his Elements (circa 300 BC).

It may be useful to note, in considering the DFTM, that the

Euclidean algorithm itself is based on the principle that the GCD of two numbers does not change if the larger number is replaced by its difference with the smaller number.

Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of

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numbers until the two numbers can become equal. When that occurs, this is the GCD of the original two numbers4.

Generally, the GCD in this case only requires one step per even number, and there are 5 trials per decimal digit. For 617 decimal digits of an RSA-2048) key, this means that breaking RSA-2048 should require less than about 600 steps in hardware when processed in parallel, with just 5 concurrent threads per decimal digit.

The GCD does not even require five times the number of decimal digits of the smaller integer. A special-purpose FPGA can reduce that number to 1 or close.

This contradicts a famous result due to Gabriel Lamé [26], who in 1844 analyzed the complexity of the Euclidean algorithm. Lamé thought that when looking for the GCD of two integers a and b, b being the smaller, the algorithm finishes in at most 5k steps, where k is the number of digits (decimal) of b, also called the b decimal-length. But Lamé did not consider FPGAs, and that a RSA key must include exactly two primes. Finding one, finds the other – immediately.

5. PERIODICITY AND QC

In finding prime numbers by exploring classical periodicity, one can use the GCD with the modular exponential (i.e., mod) function, and it reveals a periodicity on numbers.

This works numerically as follows, which is well-known (and often confused in mathematics; as no one is taking a “bad guess” here).

4 In the field of number theory, one can easily visualize the Euclidean algorithm for the GCD of two natural numbers.

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Assume that N is positive and has only two distinct prime factors: N = p1p2.

First, one needs to pick an integer a between 2 and N-1, as is well-known. One needs, also, to compute the GCD of (N,a), to the effect that a and N should be coprime.

Second, one has only to choose r to be the known number 2, which is consistent with chord oscillations in round-trips.

At this point, given some number N, knowingly the product of two primes, we are just considering that r is the number 2 (as the least possible number for round-trips).

If the decision on a is reached digit-by-digit, ranging from 2 to N-1, one has a classical (and unbearably slow) method, which is also subject to many errors (see footnote 5). By using QC, one hopes to have a collaborative approach on the decision which best a to use, where all numbers are tried in the same operation (much faster and promising less errors). But, whatever method one uses, one is measuring the same objective quantity: periodicity in a, such that a2 = 1 (mod N).

Just one final number is needed: a.

QC starts with an objective wave model using (e.g.,) the DFTM

-- propagation of waves back and forth in the chord. Following Section 3 and [18-19] for optical applications, where the equation is usually transformed into a large (e.g., 100x100) matrix equation. With these large matrices, matrix evaluation will needlessly calculate 100 modes of the 100x100 matrix, while we want only the first dominant mode. This means that computer time will be used to calculate typically less than 10 actual modes and 90 random-looking collections of points, due

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to non convergence of the process5 [18-19]. Besides, we have to store large amounts of data in random access memory, [18-19]. We can use the eigenvalue translation method [18], the revisited Prony method [19], the DFT, or other, to achieve a higher efficiency in calculating only the highest mode of the eigenequation modeling the problem. This can now be applied to the problem at hand, estimating the dominant mode of the chord oscillations (see footnote 5) -- and a.

Starting with an arbitrary function ao decomposable through the scalar product in the fp basis:

ao(s) = Σ αi.fi(s), summed over all i, where αi are in the set Q,

(e.g., taking ao(s) = 1, which is valid for any a) we construct the sequence,

ar(s) = Σ αi.αir.fi(s), summed over all i,

where the equation represents either a single-trip for the symmetric modes, or a round-trip for the asymmetric modes, exhausting all possibilities. We unroll the multiplications for faster speed, doing only additions.

We look for r=2 as the smallest integer where (a2 -1) is a multiple of N, i.e, where a2 is 1 (mod N) (e.g., following the efficient code snippet for the mod function in Fig.(3)), thus repeating ao, with a chosen propagation function [e.g.,

5 On selecting a propagation function to use in QC, like the DFT, one is affected by the Nyquist theorem on aliasing, where the propagation analysis suffers from period sub-multiple errors, due to the harmonics that are intrinsic to an interpretation of periodicity applied to the numerical mappings of periodic signals by means of sinusoids. More suitable in terms of estimating periodicity, a frequency spacing method used here, algorithmically, is the DFTM (obeying the empirical conjecture #2), which may or not use [18-19].

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18-19,21,27] (see footnote 5) -- finding the desired periodicity, e.g., by repeated subtraction6. In the lossless case, the propagation equation represents the usual DFT [21] (see warning in footnote 5). Then, since r=2, one uses the algebra identity in Eq.(1) below:

for the asymmetric modes (round-trip) to split a2 -1. Test that

is not a multiple of N, change a if needed (or, it could be left as a multiple of N and that be taken out trivially with the GCD). In that case, neither is:

a -1 (3)

a multiple of N, although their product is (in the case where the product of N is included, it can be taken out trivially with the GCD).

This is possible only if p1 and p2 are prime factors of N.

We have split a2 - 1 in Eq.(1), non-trivially, and without necessarily using one trial number at each time -- we used a collaborative effect of all possible modes, to find the period --

6 Repeated subtraction is the process of subtracting a number continuously from the large number until the remainder is found -- as zero or lesser than the actual number.

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through a7. This contradicts [26], and all known references -- we find with the DFTM that only a is needed in QC.

We have objectively considered the contribution of multiple numbers at the same time in QC. Therein lies most of the power of QC -- and avoiding costly matrix algebra, using [18-19] or the DFT [21].

The directive 'all states at once' in QC then becomes quantifiable and causal8, with no mysterious QM, ghostly superpositions, change of r, or non-causal entanglements.

As noted by Spiros Konstantogiannis [23], it is worth noting that when the modulus N is the product of two distinct primes p1,p2, there exist exactly four different solutions for the pair of integers

(ar\2-1,ar\2+1), as it can be seen by setting x= ar\2 and solving the congruence x2=1 (mod N), where N=p1p2. The existence of the previous four solutions is ensured by a well-known theorem in cybersecurity, called the Chinese Remainder Theorem, which is a basic result also in elementary number theory. Two of the four solutions are the trivial ones x=1, N-1, which are present in any case N>1.

The two trivial solutions are excluded in the analysis, since then x-1=0, which is a trivial multiple of the product of two primes, or x+1=N, which is again a multiple of the product of two primes.

7 We targeted in Eq.(1-3) an objective periodicity in the exponential function a, as a collective property of a group of numbers, considered in our QC, to find a better estimate for a – given by the period. In the lossless case, the propagation equation represents the usual DFT -- connecting both techniques of QC. Note that our technique of QC, however, does not use “imaginary numbers” or cryogenics.

8 The numbers 1 or 0 can, e.g., result from 0 + 0, depending on the carry signal.

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The other two solutions are non-trivial and include pairs (ar\2-1, ar\2+1) of integers where one integer has one of the primes and the other integer has the other prime.

In the special case where the modulus N is the product of two twin primes, i.e. of two primes that differ by 2, for instance, 3 and 5, 5 and 7, 11 and 13, 17 and 19, and so on, one of the two non-trivial solutions can be exactly the pair of the two twin primes, since then the two primes differ by 2, which is equal to the difference of the two integers ar\2+1, ar\2-1 and for this reason such a solution is possible.

The use of twin primes is to be avoided in cryptography. This is well-known to have been exploited recently, to break the encryption algorithm RSA. Therefore, one should sufficiently randomize the two prime numbers, when used to generate stronger RSA keys.

One can also create a plane9. Without assigning any numerical meaning to √-1, one can visualize √-1 as a simple 180 degree rotation, not as an imaginary number or as an irrational number, and use it effectively in mathematics and QM, without contradictions [cf. 1] such as i = -i. This creates the set Q* and deprecates the sets C and G (to be published elsewhere).

6. QC

In viewing prime numbers as not about their effectiveness on factoring (they must, as a theorem of arithmetic), but as two

9 This leads us to consider the famous phrase "a negative times a negative equals a positive" as an easily accessible truth, explored in ResearchGate online discussions. We did this by considering a negative number as a 180 rotation, so their product is a 360 degree rotation, back to positive. One cannot achieve the same ease of visualization by considering a negative number as a "loss", as done in the major movie "Stand and Deliver".

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