This question has been closed, reaching a YES conclusion and preventing being hijacked by 'wolves" in ResearchGate. It served the purpose of explanation to those interested, as an open group, from a core of 10 people who participate behind the scenes. Enjoy and pursue the new ideas with your contributions in your space. The question is now open again.
To be objective, K=26 661 462 837 357 923.
This large base 10 number is found to be the product of two prime numbers, K=p×q.This breaks the RSA cybersecurity method for K.
The much larger number H=74 481 443.869 551 262 986 707 503 438 165 513 011 429 940 762 703 277 812 267 530 769 921 052 121 342 275 484 565 273 568 067 051 66*10^991 with the missing integer values known, albeit not shown here, reveals quantum properties of numbers, that help break RSA structurally, making it impossible to protect.
That number is found quickly to be the product of two very large prime numbers, H=m×n, where m=2189435657951002049032270810436111915218750169457857275418378508356311569473822406785779581304570826199205758922472595366415651620520158737919845877408325291052446903888118841237643411919510455053466586162432719401971139098455367272785370993456298555867193697740700037004307837589974206767840169672078462806292290321071616698672605489884455142571939854994489395944960640451323621402659861930732493697704776060676806701764916694030348199618814556251955925669188308255149429475965372748456246288242345265977897377408964665539924359287862125159674832209760295056966999272846705637471375330192483135870761254126834158601294475660114554207495899525635430682886346310849656506827715529962567908452357025521 and n= 340185579782030309029142285845485748073406778702270938755484147318382420338087834406828955714187005654640257038495796545155402280055987076251704557994637589726712709889312042801858044039590155407650471667907995888292123909278046563998441725881316702608454953284969473141146885140822683049274853701491, breaking RSA with values naively considered large enough to be "safe".
We postulate without proof here, except numerical, that this exemplifies how RSA can be quickly broken, e.g., for a 2048 bit-length number. in 2048 bits one can store a number with 617 decimal digits; and we passed that in the last example. The larger the number of digits in each prime number, the easier it is to numerically calculate them.
RSA gets weaker with large prime numbers. This is a structural weakness, much more important for cybersecurity than numerically finding prime numbers.
This shows objectively the weakness of RSA. QM is our most successful model of nature. Classically, i.e., without QM, those results are not calculable and RSA looks stronger for large numbers.
RSA seems to be broken easily by quantum computing -- more so for very large numbers. It is a hopeless case using QM, and quantum computing.
This shows the importance of periodic structures in mathematics. And we can find them using QM, and quantum computing.
What is your opinion?
It is elementary. One can only observe and be grateful. The calculations did not have to use Linux and the absolute-precision program bc; but were done using a Samsung cellphone using Android.
Quantum computing in a cellphone! This opens the market.wide. No cryogenic behemoths. No "software as a service". No remote calculation.
Quantum computing can be done in a ... watch. Step aside doubters. Who doubts just looks stupid.
To comment, I exclude irrational numbers from any rational model in nature, including all arts, skills, and sciences.
Once that is done, many nebulous things get clear and computable in short time. QM is our most successful model of nature.
Any prejudice against QM just needs time to be clarified. Nine mothers cannot make a baby in 1 month -- time is a factor.
This question benefits from noisy inputs in another question.
Continuity is denied in mathematics because irrational numbers form an empty set. Continuity is physically denied because of Quantum Mechanics (QM). QM is our most successful model of nature, in any art, skill, or science. QM applies squarely to mathematics and this question and answer proves it.
To attentive readers, I did not say that all digits are not known. They were.
The trap for "wolves" , inspired by the Almighty beforehand, was to write the decimal equivalent of the product, omitting the later digits, which were trivial -- particularly the last one. Must be 1. Nothing new.
But those who wish to find faults deserve to find them, and get their return, which is educative (not punitive, as only wolves fall in the trap for wolves), and motivates them to evolve.
I sincerely wish the "wolves" in ResearchGate to become sheep, non-aggressive. It will happen; it is a matter of time. Everything evolves. We all must! Toward the Creator. Wolves become shhep, and beyond.
Prime numbers obey a quantum property named #4, characterized by irreducibility.
This is exemplified in the largest prime number given above, as m=2189435657951002049032270810436111915218750169457857275418378508356311569473822406785779581304570826199205758922472595366415651620520158737919845877408325291052446903888118841237643411919510455053466586162432719401971139098455367272785370993456298555867193697740700037004307837589974206767840169672078462806292290321071616698672605489884455142571939854994489395944960640451323621402659861930732493697704776060676806701764916694030348199618814556251955925669188308255149429475965372748456246288242345265977897377408964665539924359287862125159674832209760295056966999272846705637471375330192483135870761254126834158601294475660114554207495899525635430682886346310849656506827715529962567908452357025521, where every number in m is immutable, in order for the entire m to stay as a prime number. Thus, every prime number corresponds to a coherent structure in mathematics, reflecting QM properties. QM is our most complete model of nature, and applies to any art, skil, or science-- including physics and mathematics.
Another number arrangement that corresponds to a coherent structure in mathematics, are the digits of pi. Might it be interesting to compare the two in terms of periodicity or quantum properties?
We are led to this question because we know that QM is our most successful model of nature, and applies to mathematics and physics. Could we find quantum properties in the digits of pi? Of a prime number?
This thread shows the inadequacy of using Peano axioms in mathematics, to define numbers. The quantum properties are not transmitted thereby.
We are left with the well-known observation by Leopold Kronecker (1823-1891), that "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk".
The Almighty included the quantum properties in mathematics, from numbers, and humans cannot undo them.
This question has been closed, reaching a YES conclusion and preventing being hijacked by 'wolves" in ResearchGate. It is now open again.
It serves rhe purpose of explanation to those interested, as an open group. Enjoy and pursue the new ideas with your contributions in your space.
As published in ResearchGate, here is the solution to break RSA-2048 and even higher, fast.
In the old method, in order to #factor one would need to inchworm, prime number by prime number, to reach 308 decimal-digits.
That would take some trillion years in computation. Impossible?
Now, QC reveals that prime numbers are #eigenvalues. Must obey the Schrödinger equation for bound-states. That is a diophantine equation -- solutions ARE only integers with arbitrary-length, as published in Phys. Rev. Lett. in 1982. That's valuable, including for structural dynamicists.
One can calculate any 308 decimal-digit prime number, as one already knows and does it, and then just add or subtract even numbers and test -- in a quick search one hits the jackpot!
We have found the only one prime number that factors RSA.
Done -- RSA-2048 has been broken, by the Prime Function Theorem (Gauss, +220 years ago -- see image -- from ). The world has to change.