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types of partition, one of them objective, we can more easily bring in QM and QC in objective and useful (to everyone) terms.

Here, one can use an analogy between primality testing and the resonant condition in physics, so that solving the latter solves the first.

This is similar to the well-known Schor's algorithm for quantum factoring, although less laborious, and not using complex numbers (that we consider fictional in the set Q, cf. [1]). This can provide primality testing in mathematics, by the sole use of objective, physical laws [1]. The resonant condition for two prime numbers leads to N = p1p2 = p.q, where multiples of p and q are avoided naturally, and is in the same form as a RSA public key.

QC becomes one consequence of opening up the "black box" of QM, described in previous results published by the author

[cf. 1, 22]. QM is shown numerically to be exact to more than 101000 decimal digits [16]. QM, therefore, is our most rigorous model of nature!

To be able to use the resonant condition in physics, we need to not consider infinitesimal or irrational differences in frequencies.

This means that we consider the material conjecture #2, and no Problem of Closure [1].

We consider only rational numbers and can differentiate discontinuous functions (not just distributions) -- deprecating most textbooks in analysis [cf. 1]. A new era in mathematics can start.

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Furthermore, taking advantage of the well-known 1:1 relationship between rational numbers and positive natural numbers (set N), we are led to consider the set N.

In that, similar to computers, we work only with the Boolean set

B={0,1}, and build Bn as is well-known. Thus, imaginary numbers, infinitesimal numbers, and p-adic numbers are not included (i.e., our empirical conjecture #2).

Now, in the Boolean set:

offering instant calculations.

Thus, QC can work fast with 'all states at once', which follows non-Boolean algebra, with even a googol of states. We note that, just considering three logical states, one has many more combinations than with Boolean algebra [22]; for example,

19,683 compared with 16 for binary operators. With any number of logical states, we call that simultaneous multiple-states logic (following ‘all states at once’), the basis of

QC, together with collaborative calculations.

By following a careful 1:1 path from the sets R and C, to N+ and then to the set B={0,1}, where there is no actual calculation by software for any expression, one can easily implement QC. The negative values are handled as explained in Section 8.

Thus, a difficult problem in the set R (or C) is transformed into a trivial problem in the set B, considering QC.

The equivalence of the techniques has been hidden for about 2,500 years – since Pythagoras. One considered illusions, fictional entities, as to “exist”, with imaginary numbers, irrational

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numbers and infinitesimal numbers appearing in set Q. Instead, we considered our empirical conjecture #2.

We take the position of a digital computer and interpret “to exist” to mean that in a practical sense -- to exist is “it can be constructed from the set B = {0,1}”.

To “exist”, then, is to be calculable numerically by digital hardware. This can happen in a digital computer, or a Field Programmable Array (FPGA), using binary logic with B={0,1} and QC, or three-state logic with A={0,1,X} classically. See Section 2.

Our empirical conjecture #5 is that nature is solving problems only numerically, by finite difference schemes using the set N.

The basis of this conjecture #5 is nature itself. We proposed such solutions to calculate bound states using the Schrödinger equation using the set Q [27] (that we have later re-cast as a difference equation for the curvature [cf. 27]), discovering QC in 1982 [27] -- heralded by an observation 40 years before [27].

This has been used by A. Gerck and E. Gerck in 2023 [29] to confirm most results in Newton’s and Leibiniz’s calculus, while invalidating some. We adopted the calculus by Mādhava, discovered 250 years before Newton -- and so far ignored, even though Newton’s and Leibiniz’s calculus produce no results in the case of discontinuous functions [1]. Mādhava calculus allows discontinuous functions [1], satisfies the Schrödinger equation [cf. 27], opens new results in QM [27], and creates the conditions for discovering QC and using it in calculus [1, 27-28].

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The Big Bang, which may not have been the first, is our empirical conjecture #6, that “There are multiple Big Bangs”. A Big Bang is revealed then as an expression of calculus.

Furthermore, we consider that calculus, SR, QM, GR+QM, and

QC already existed in nature in multiple Big Bangs! These things have existed since time immemorial, and support our empirical conjecture #5. Galaxies were then created based on calculus. Nature is exact, rigorous in spacetime. We reject the QM interpretation by Heisenberg. We explain that by difference equations using the set Q, as done in [28] without infinitesimals, or approximations.

Calculus optimizes understanding, as the tool that allows one (i.e. also particles) to prevent trial and error in finding the exact result of optimization problems [28].

Some may claim, without proof, that "a completely rigorous definition of infinitesimals can be used to derive most (if not all) of and calculus and modern atoms) analysis" (but no one can name an infinitesimal numerically). Calculus should use no infinitesimals or irrational numbers, as a consequence of QM and QC. This was our empirical conjecture #2.

7. GCD OPTIMIZATION

In the GCD of a and b, one takes b to be smaller than a, and both belong to the set N.

The Pascal code segment in

Fig.(2) on the right exemplifies a well-known way to calculate the GCD as done usually.

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The code snippet in Fig. 3 below is much faster to implement, with no mod calculation. The faster speed is explained by the well-known theory of computational complexity, which gives the time required as a function of N.

The code for the GCD can be optimized, but note that there is no such thing as subtraction in the realms of Boolean mathematics and digital hardware (e.g., using bignum).

To subtract in bignum in Figs.(3) (and any other arithmetic operation in bignum, such as multiplication, exponentiation, logarithm, module, and division) one uses encoding and addition -- which can be done natively (fast) in digital hardware and uses Boolean mathematics.

In the commercial cellphone used by us in 2023, bignum was supported in more than 101000 decimal digits, more than sufficient for 2048 binary digits (using approximately 617 decimal digits).

Particularly in subtraction as in Fjg.(3), one first finds the 2's complement of the subtrahend. In step-2, one adds the first number and 2's complement of the subtrahend. In step-3, if a carry is produced, one discards the carry and outputs the

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result. If there is no carry then one takes the 2's complement of the result to output.

Subtraction as an operation would imply the existence of negative numbers: 5 - 3 is the same thing as 5 + (-3), and in the

Boolean set (the set B), negative quantities are forbidden. This is similar to forbidding imaginary numbers, infinitesimal numbers, irrational numbers, and p-adic numbers in set Q [1], and they are also forbidden in set B.

The actual calculation of Fig.(3) is implemented as a binary calculation, which is done using only digital hardware -- one number at a time, although it can be done concurrently – albeit not using QC.

A binary number such as using N values of the set {1,2,3, …,N-1} is 1:1 equivalent to using only the B set {0,1} also with N values in BN. Further, we can calculate subtraction using addition (with 2-complements) using digital hardware operations. Also, 0n = 0 and 1n = 1, with immediate results.

Other techniques, such as the General Number Field Sieve, are well-known to be used to crack numbers approaching 800 bits in practical terms today.

The potential influence of radiation sources (e.g., cosmic rays) were accounted for by running the results several times, with zero spread. The results are exact, which can be easily verified by third-parties. All results shown were measured in Sonora, Calif., U.S., near Yosemite.

We do not provide the source code, for the aforementioned cybersecurity reason, but relate the numerical results in Section 9, with QC as well.

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8. ASYMPTOTIC APPROXIMATIONS

The prime number theorem states that the prime numbers become less common as their length increases.

This makes no hypothesis on an objective model.

How primes are distributed among positive integers is described asymptotically by various approximations -- without any hypothesis on the behavior. It is also known10 that if p1 > p2, then p1 = (2m+2) + p2, for some m≥0 in the set Z, just not if p2 =

2.

Our empirical conjecture #7 is that “The larger p1 or p2 as prime numbers, except 2, the further apart are p1 = (2m+2) + p2, for any m≥0 in the set Z”.

This confirms the well-knowntwin prime conjecture, which states that there are infinitely many primes p such that p + 2 is also prime, but also we conjecture them to be more rarified when p is increased.

Then, as a consequence of the prime number theorem, one gets various asymptotic approximations for the nth prime number, beginning with pn n log n [24], which can be written as e(pn)/n = n.

A better approximation is given in [25]. In 1999, P. Dusart proved that, for k in the set N, the kth prime is greater than k(\ln k + \ln\ln k-1) for k ≥ 2.

Not as an approximation but exactly, we make an empirical conjecture #8, without proof here, that an exact formula

10 All prime numbers except 2, are odd numbers.

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describing how prime numbers are distributed among positive integers can be described rigorously to any such number.

We do not provide the expression for empirical conjecture #8, for cybersecurity reasons.

9. QC NUMERICAL RESULTS

RSA 2048 refers to the key size used in RSA encryption. The key size supposedly determines the strength of the encryption and the level of security it provides. RSA 2048 uses a 2048-bit key, making it supposedly significantly more secure than smaller key sizes like RSA 1024.

We contradict this speculation numerically with QC, using a

2048 key given by RSA and yet unbroken. This challenge dates to at least 2007 [30]. The page [30] serves as an archive for the factoring challenges conducted by RSA Laboratories through 2007. The key numbers will be published elsewhere, with a delay, for security reasons.

RSA-2048 has been particularly sought after, to break the encryption -- and RSA declared at time: “The RSA Challenge numbers are the kind we believe to be the hardest to factor; these numbers should be particularly challenging. These are the kind of numbers used in devising secure RSA cryptosystems.”

The numerical results, which third parties can use to verify, in simple multiplication to break RSA-2048, are given elsewhere.

This confirms QC; many applications of QC in other areas are being pursued, including in healthcare [31].

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Periodicity estimates, as the heart of any QC, may be improved. Biological sequences , e.g, are rarely exactly periodic and the estimated period returned by exploratory analysis may be anywhere from very weakly to very strongly dominant with respect to the remaining sequence components

(which may contain other periods or be essentially non-periodic) [32].

10. CONCLUSION

RSA is pathologically broken. Increasing the key size is no help, and may make it easier to decrypt.

The U.S. NIST has speculated that 2048 bit keys will be valid up to about the year 2030. It just now (2023) has been broken in seconds, using a commercial cellphone and QC.

A post-quantum, HIPAA compliant, end-to-end, patent-free, export-free, secure online solution, is being created based on ZSentry used from 2004 to 2014, to replace RSA. One needs a quantum-resistant algorithm, because all existing public-key encryption can be broken. This will be pursued publicly soon as an IETF RFC, and as a NIST submission.

We further presented 8 empirical conjectures, all supported by experiments, that should be helpful in the further development of QC and QM.

This work may kindle other applications of QC and QM, which are confirmed based on this study.

A medical doctor, e.g., can use QC in his mind while treating classically for a pathology, using available means. GR is to be unified with QM following QC. Cosmology can consider that

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calculus, SR, QM, GR+QM, and QC already exist ontologically, in nature.

Additional work is suggested on periodicity estimation (see footnote 5).

FUNDING

This research received external funding from Network Manifold Associates, Inc. (NMA), Safevote, Inc, and PlanaltoResearch. The authors declare no conflict of interest.

ACKNOWLEDGMENTS

The authors are indebted to Spiros Konstantogiannis, the late Mādhava of Sangamagrāma, and six anonymous reviewers. ZSentry development from 2004 to 2014, had the benefit of dozens of contributions, including by the many angel investors in NMA, principally with the late Einar Stefferud, Graham Tanaka, Kurt Neumann, Michael Norden, and Vernon Neppe. The results shown in this manuscript, include the contribution of several individuals -- some of which the author does not know personally, and some of which may come spontaneously online. Repeated updates are suggested, so as not to use any outdated terms. Any suggestions are always welcome! This is a new model for an online manuscript, where anyone may collaborate, much like a question. We answer two questions: what role can QC have, and what it may reveal. To participate send me a private or public message, by ResearchGate or email (see address above). ResearchGate discussions and private messages were also used, for “live” feedback, important due to the physical isolation caused by COVID.

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