Theorema Magnum MCMLI: der Vergleichssatz von Rauch

Kommentare (16)

1. #1 frbrn

   5. November 2020

   Über den “Skandal” hätte ich gern mehr erfahren. Wikipedia (en/de) hat nur Blaschkes Nazi-Vergangenheit. Eventuell auch Hinweis auf ein Paper/Zeitungbericht…

2. #2 Thilo

   5. November 2020

   Wenn ich mich richtig erinnere, habe ich das aus einem Buch von Reinhard Siegmund-Schultze (Mathematiker auf der Flucht vor Hitler. Quellen und Studien zur Emigration einer Wissenschaft, Vieweg: Braunschweig 1998), einer Sammlung von Originaldokumenten, bspw. Briefwechseln des Springer-Verlages mit verschiedenen Mathematikern. Ich habe das Buch jetzt nicht vorliegen, insofern kann ich gerade nicht nachprüfen, ob das tatsächlich von dort ist.

3. #3 frbrn

   5. November 2020

   Herzlichen Danke für die schnelle Antwort!

4. #4 Marc Nardmann

   6. November 2020

   “Cartan hatte bewiesen, dass Riemannsche Mannigfaltigkeiten durch ihren Krümmungstensor lokal eindeutig bis auf Isometrie festgelegt sind.” Falls ich diese Aussage richtig interpretiere, stimmt sie nicht (ohne zusätzliche Voraussetzungen). Dieses Problem wird z.B. in Bergers Buch “A panoramic view of Riemannian geometry” diskutiert, in Abschnitt 4.5.2 (S. 235-236) und Abschnitt 6.3.1 (S. 272-273). Die Gegenbeispiele sind allerdings nicht offensichtlich.

5. #5 Thilo

   6. November 2020

   Gemeint ist folgendes Statement: wenn man zwei Metriken hat, in einem Punkt die beiden Tangentialräume mittels einer Isometrie identifiziert und dann jeweils zu jeder Geodäte durch den Punkt und zu jedem Vektor X in der einen Riemannschen Mannigfaltigkeit denjenigen Vektor X’ in der anderen Riemannschen Mannigfaltigkeit betrachtet, der das Bild desselben Vektors unter der Parallelverschiebung entlang der entsprechenden Geodäten ist; wenn dann für Vektoren X,Y,Z,W der Krümmungstensor jeweils mit dem von X’,Y’,Z’,W’ übereinstimmt (für alle Geodäten durch den Punkt), dann müssen die Metriken lokal isometrisch sein. Das findet man z.B. in Cheeger-Ebin, nach meiner Erinnerung ist der Beweis auch nicht besonders schwierig. (Auch hier: ich habe gerade keinen Zugriff auf das Buch. Eine Online-Quelle wäre die Vorlesung 8 in http://www.mathematik.uni-muenchen.de/~tvogel/Vorlesungen/Riemann/skript-Riemann.pdf )

6. #6 Marc Nardmann

   6. November 2020

   Dieses Statement ist wahr. Aber naiverweise würde man doch unter der Aussage “Riemannsche Mannigfaltigkeiten sind bis auf Isometrie lokal eindeutig durch ihren Krümmungstensor festgelegt.” folgendes verstehen: “Wenn f: M -> N ein Diffeomorphismus zwischen Riemannschen Mannigfaltigkeiten ist, sodass die von f induzierte Abbildung dem Krümmungstensor von M den Krümmungstensor von N zuordnet, dann ist f eine lokale Isometrie.” (Oder zumindest: “… dann sind M und N lokal isometrisch.”) Ich wollte nur darauf hinweisen, dass das (beides) im Allgemeinen nicht stimmt. Nun ist ja alles geklärt.

7. #7 Nikolaus Graf zu Castell-Castell

   Prague (CR) und Zug (CH)

   6. November 2020

   Nikolaus Graf zu Castell-Castell Varsavska 36 CZ – 12002 Prague
    
   UNITED STATES PATENT AND TRADEMARK OFFICE P.O.Box 1450 ALEXANDRIA, V.A. 22313-1450

   Provisional Application for patent acc. 35 USC par. 111(b) SMALL ENTITY

   01. 06. 2020

   The factoring of large integers by the novel Castell-Fact-Algorithm, 12th part,
   to crash the RSA codes.
   Following two of three ways to hack the RSA Code by the novel Tietken-Castell-Prime-Algorithm in order to definitely and correctly identifying and generating prime numbers of unlimited size in an indirect way.
    

   Abstract
   Our essays 1 to 11 describe the applicable Castell-Fact-Algorithm, which factorizes large integers, was ignored and rejected by economy and politics.
   Innovations concerning data protection and security seem not to be in great demand by neither the NSA, nor by Facebook & Co.
   In the following essay part 12, we introduce the
   Tietken-Castell-Prime-Algorithm, which is able to indirectly
   a) produce prime numbers
   b) identify prime numbers,
   c) and which is suitable for the factoring of large numbers (composed of prime numbers) after creating a comprehensive registry of numbers.
    

   Objective
   Following question would not appear, if for the factoring of large integers, large integers were given, which reliably emerged from the multiplication of two prime numbers.
   It is debatable, whether the given “large number” is a product of two prime factors or a prime number itself (which can not be factored). Maybe it is even only product which ends with the digits 1, 3, 7 or 9. Therefore, a reliable method for clarification is needed in order to determine what exactly the given large number represents.
   Taking a look into the many existing, non-reliable and ambiguously functioning methods of solving the issue, one would realize that a new approach without uncertain multiple trials and error rate (Fermat, Miller-Rabin, Chinese remainder theorem, Mersenne prime, etc.), as well as without Probability Theory, approximation, repeats, etc., is needed in order to obtain more than just a “most likely prime”.
   Even RSA (Ron Rivest, Shamir, Adleman) only utilizes “random numbers”, which do not generate exact prime numbers but only “probable prime numbers”.
   The idea of the novel “Tietken-Castell-prime-Algorithm” saves the user the aforementioned uncertainties and ambiguities.
   It delivers 100 % prime numbers, which are definite and in all magnitudes.
   It furthermore avoids the disadvantage, that big anterograde “skips” cause smaller prime numbers which are in between these skips to be overseen or omitted, which is the inconvenience of the RSA-multiplication of two identical prime numbers.
   Shortly, it creates a dearrangement or disarray in the assumption of which other prime numbers do in fact exist.
   If there would be total clearness and transparency, the RSA would not have multiply “probable prime numbers” with themselves for reasons of certainty.
    

   Personal preliminary remark concerning the “Tiekten-Castell-Prime-Algorithm”
   The hereby presented novel algorithm is not only able to identify prime numbers in an indirect fashion, but also to generate them in that same fashion.
   The solution suggested by the aforementioned novel procedure is simple and self-evident.
   To generate a prime number, large enough to fill a dozen of folders, may be an impressive display of the quantitative potentials of current computers , but has nothing to do with the ever-growing qualitative human thinking (comparison to Gimps and the 400 years old Mersenne prime, etc.), apart from initial thoughts of an algorithm for the relatively small prime numbers in the beginning.
   It is nothing but a currently still useless show with skyrocketing electricity bills and only a minimum of human in-house effort.
    

   Verbal explanation of the “Tiekten-Castell-Prime-Algorithm”
   The fundamental consideration of the “Tiekten-Castell-Prime-Algorithm” is the currently still established impossibility to rapidly detect and generate large prime numbers, through an indirect manner.
   a)
   If prime numbers are not simply and correctly producible, one can use so called Pendants, which emerge from the differential method and are ideally (as shown here) exactly calculable.
   These are the well-ordered, constant-remaining counting factors on one side and the two-step cardinally-growing counted factors of the also well-ordered and comprehensively growing non-prime numbers (or products in the close vicinity of the wanted prime numbers) on the other side.
   The prime numbers in this case are the „remainders”. They are numbers without factors!
   b)
   If prime numbers can be generated in this indirect fashion, then in a way that these prime numbers are identifiable accroding to their position.
   Due to simplicity and the fact that all steps and procedures stay constant, there is an aligned order of prime numbers built up by their respective values, which exhibits no possible gaps or mistakes; meaning every prime number is captured and made visible.
   Once put into motion, the Tietken-Castell-Prime-Algorithm is able to build up a constantly growing registry without any following input, which is ordered according to the scores of numbers and can be accessed at any time.
   Thereby, no human interaction or effort is needed – the procedure of operation is automatized.

    
   Numerical examples of the “Tiekten-Castell-Prime-Algorithm”
   The algorithm builds itself up only with the final digits 1, 3, (5), 7 and 9 per decade (every 10th).
   Number “5” is put in brackets, because it acts as an exception.
   It may (except as a single digit) never become a prime number, even with added decimal digits, due to the fact that numbers with the final digit “-5” are always divisible by 5.
   However, the Tietken-Castell-System counts the “5” as well, in order to keep a consistent increment and to avoid the necessity of a double-step every time the final digit “-5” comes up, in which the “5” will be skipped.
   The following graph shows how the Tietken-Castell-Registry grows in a constant-remaining fashion (score-wise). According to our premise of a constant-remaining decimal system, it could be continued indefinitely.
    
   a)
   In each of the indefinite amount of rows, only numbers with the final digits 1, 3, 5, 7 or 9 are being captured, because every prime number has to exhibit either 1, 3, 7 or 9 as a final digit (digit 5, as exception, is dropped).
   However not every number with these final digits is a prime number.
   So, in order to comprehensively capture all prime numbers without gaps, all numbers which exhibit one of the four prime-end-digits, have to be included into the registry as well. Among them also the prime numbers are to be found. These are identified by the fact that there are no existent factors for them.
    
   b)
   The rows within the Tiekten-Castell-Registry always look alike. Each row exhibits the final digits 1, 3, 5, 7 and 9.
   Every newly added row, receives an additional decade with 1, 3, (5), 7, 9 on the end.
   1 3 5 7 9
   11 13 15 17 19
   21 23 25 27 29
   31 33 35 37 39
   41 43 45 47 49
   51 53 55 57 59
   61 63 65 67 69
   71 73 75 77 79
   etc.
   or:
   1001 1003 1005 1007 1009
   1011 1013 1015 1017 1019
   1021 1023 1025 1027 1029
   etc.
   or:
   2381 2383 2385 2387 2389
   2391 2393 2395 2397 2399
   2401 2403 2405 2407 2409
   2411 2413 2415 2417 2419
   etc.
    
   c)
   The claim here is that there is no limitation of numbers towards the top!
   However, not an approximately 2-million-digits prime number is looked for, but preliminary “only” “large numbers” of 1000- to 2000 digits, which emerged through multiplication of two prime numbers.
   According to our estimation, this task (even with the consideration that prime numbers are getting rarer and rarer within the increasing numbers) could be executed easier and faster than going just by the vague procedure of finding prime numbers by chance, which are not even reliably identified or confirmed.
   In order to do the task correctly the “Tiekten-Castell-Prime-Algorithm” has to count up without gaps, constantly add, as well as keep the sustained connection to each previous factor.
   Furthermore it has to establish multiplications between two prime numbers and save the results as „large numbers” to recall them later with the respectively given factors!
    
   d)
   In order to only keep numbers with the final digits 1, 3, 7 or 9 in the registry, all numbers with the final digits 0, 2, 4, 6 or 8 have to be excluded from the registry.
   Final digit “5” is counted along in the registry, but plays no further role.
    
   e)
   Each number of the register functions and provides itself as a constant factor for the own respective row of following numbers (in distances of two times the respective number).
   Every time a prime number emerges due to the lack of potential factors, it is also multiplied by itself and starts a new unlimited row of numbers in the registry, from the point of the emerged product (a large number from two prime numbers).
   Prime numbers are numbers which occur without factors within the Tietken-Castell-Registry.
   However, if there is an involvement of factors (two or more) in the formation of the given number, it is impossible to be a prime number.
   Because the respective factors are past on the “left side”, the product is calculated there already.
   If this in the past calculated number shows up then again, it will already be known as the product together with its respective factors.
   This way, it can also be determined whether the given number is a “large number”, meaning a product of two prime numbers.
    
   f)
   Following, the mechanism of action of numbers within the Tietken-Castell-Algorithm is exemplified.
   Inevitably the origin starts with small numbers. Though, the principle of approach and utilization continues, as before indefinitely towards the top, because the laws of the decimal system remain the same for small and large numbers. Also the constant approach of the
   Tietken-Castell-Prime-Algorithm remains unchanged.
    
   g)
   1. Row, 1. Number: 3
   3 is (as all numbers of the registry) multiplied, makes 9 and begins the first row of numbers from here, which traverses through all, constantly increasing numbers in the registry.
   This number 3 remains as a counting constant and builds up new numbers; first with itself and then with the uneven numbers (5), 7, 9, 11, 13, (15), 17, 19, 21, 23, (25), 27, 29 31, 33,(35), 37, 39 etc., which sometimes may appear as normal products, but in other cases also happen to be products of two prime numbers.
   3 is a prime number, because it has no factors “from the left” in the registry.
   If 3 is multiplied by e.g. 11 or 13 (for which the same rules apply), the respectively emerged product of 33 or 39 acts as a “large number” suitable for encrypting.
   With the increasingly frequent rows of numbers, which run indefinitely through the entire register, there can be multiple assignments of numbers, in places where two rows of numbers overlap. This means that the respective number has multiple factors.
   Though, most importantly for the Tietken-Castell-Prime-Algorithm in this context, is to determine whether the number even has factors or not.
   If so, it can be elicited, whether these were prime-factors and if the product is suitable as a
   “large number” for Cryptography.
   If the number has no factors, it is (as previously mentioned) a prime number itself. In which case there can not be a multiple assignment.
    
   h)
   (1. Row, 2. Number: 5
   5 is a prime number, similar to number 3, because there are no possible factors that could lead to it as result.
   Its row of numbers starts from 5 * 5 = 25, but even with added decimal digits it will never be a prime number again, due the fact that it is always divisible by 5).
    
   i)
   1. Row, 3. Number: 7
   7 is as well a prime number, starting its unlimited march through the Tietken-Castell-Registry with 7 * 7 ( = 49) as a 7-row, then 7*9, 7*11, 7*13, (7*15), 7*17, 7*19 etc.
    
   j)
   1. Row, 4. Number: 9
   Number 9 acts as an interstage of the 3-row and therefore not a prime number.
   However, together with decimal digits, it builds an important prime-end-digit and often results in prime numbers itself (e.g. 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229 239, 269, 349, 359, 379, 389, up to infinity!)
   The 9-row, which will traverse the registry starts with 9 * 9 ( = 81).
   Together with number 9 as a constant factor and the other uneven factors which stay constant in all rows (besides 9, the 11, 13, 15, 17, 19 and so on, indefinitely), the first numbers of the 9-row a built-up.
   The fact that the single digit “9” is not a prime number, prevents it from potentially creating
   “large numbers” used for Cryptography, even though the counted second factors 11, 13, 17, 19, 23, 29 and 31 are prime numbers and would have been suitable for it.
   This initial situation though, changes its context with decimal digits. Already from the aforementioned 19, 29, 59, 79, 89, 109 etc. on, the prime-end-digit 9 is back in the race as a part of prime numbers.
    
   k)
   Résumé of the 1. Row:
   Already with these four mentioned four (exactement three) examples (3, (5), 7, 9 only, it is visible that the factors (given they exist) follow a regular order.
   On one side are numbers which always have the same final digits in the same order (1, 3, (5), 7, 9), while on the other side the same uneven numbers are constantly and cardinally counted-up in double steps.
    
   l)
   2. Row, 1. – 5. Number: 11-9
   The second row follows the same fashion of calculation as the first one (same as all indefinite, following numbers).
   (1)
   11 is multiplied by 11, 13, (15), 17, 19, 21 etc.
   After multiplication by itself, 11 * 11 = 121 ( a “large number”), it continues its row with 11 * 13,
   (11 * 15), 11 * 17, 11 * 19, 11 * 21 etc., up to infinity.
   For the purpose of encryption, it would not be necessary to multiply a prime number like 11 by itself, because the other prime numbers to form a “large number” with are known.
   (2)
   13 is multiplied by 13, (15), 17, 19, 21, 23 etc.
   After multiplication by itself, 13 * 13 = 169 ( a “large number”), it continues its row with (13 * 15), 13 * 17, 13 * 19, 13 * 21, 13 * 23 etc., up to infinity.
   For the purpose of encryption, it would not be necessary to multiply a prime number like 13 by itself, because the other prime numbers to form a “large number” with are known.
   (3)
   (15 is multiplied by 15, and after by 17, 19, 21, 23 etc., but could never create a “large number” for the RSA-Cryptography, due to the fact of only being a prime number as a single digit “5”).
   (4)
   17 is multiplied by 17 and 19, 21, 23, (25), 27, 29 etc.
   After multiplication by itself, 17 * 17 = 289 ( a “large number”), it continues its row with 17 * 19, 17 * 21, 17 * 23, (17 * 25), 17*27 etc., up to infinity.
   For the purpose of encryption, it would not be necessary to multiply a prime number like 17 by itself, because the other prime numbers to form a “large number” with are known.
   (5)
   19 is multiplied by 19 and after by 21, 23, (25), 27, 29, 31, 33 etc.
   After multiplication by itself, 19 * 19 = 361 ( a “large number”), it continues its row with 19 * 21, 19 * 23, (19 * 25), 19 * 27, 19 * 29 etc., up to infinity.
   For the purpose of encryption, it would not be necessary to multiply a prime number like 19 by itself, because the other prime numbers to form a “large number” with are known.
   (6)
   21 is multiplied by 21, then 23, (25), 27, 29, 31, 33, (35), 37 etc.
   After multiplication by itself, 21 * 21 = 441 ( a “large number”), it continues its row with 21 * 23, (21 * 25), 21 * 27, 21 * 29, 21 * 31 etc., up to infinity.
    
   m)
   Résumé of the first to the indefinite “last” Row:
   The above shown factors prove the order, predictability and correctness of this Tietken-Castell-Prime-Algorithm.
   None of the previously mentioned unreliable and complicated methods were utilized in this algorithm for determenitation, whether a more or less randomly found- or unreliably calculated number is in fact a prime number.
   If in one row, here for example the third row, where 21 and 27 can be seen, we know from the first row that 21 belongs to the 3-Row (from 3 * 7 or written in form of addition to 3 + 6 + 6+ 6), as well as that 27 belongs to the 3-Row (from 3 * 9 or in form of addition 9 + 18).
   However, no factors lead to the numbers 23 and 29 of the same (here third) row.
    
   n)
   The Distances
   It might be a relief for the algorithm, to only having the necessity to perform addition.
   It is observed that the distances between all numbers in a row stay constant, due to the first multiplication by itself (meaning a multiplication by 2). It always builds exactly twice the first counting factor.
   Insofar, the algorithm could (after the initial multiplication) build the following numbers in a row also via addition of the ever constant distances.
   (1) 3 to 9 builds the distance 6; therefore the following numbers after this 9 are 15 (=9+6), 21 (=15+6), 27 (21+6), 33, 39, 45, 51, 57, 63, 69, 75 and so on.
   ((2) 5 enlarges by 10.-steps (from 2 * 5)).
   (3) The same principle accounts for number 7, where distances between the numbers in that row will be 2 * 7 = 14 (49, 63, 77, 91, 105, 119, and so on).

    
   The beginning of the 3-Row:
    
   In the following graph, the factors of all numbers are to be found on the right hand side, which are hinting towards the later product. The algorithm saves and stores these.
   On the left hand side of those later products, their respective factors are noted once more in order to demonstrate the connection between them two.
   As now known, prime numbers do not have such hints towards the factors on their left hand side.
   Nevertheless, the absence of these left hand standing factors next to the numbers in this partial list (which only shows the progression of 3; so it is incomplete), does not give information about which kind of number we are looking at (prime number, “large number” or simple product).
   Again, there are several multiple assignments of single numbers in the beginning of the registry, which means that rows are overlapping.
   For example: crossing of the 3-Row with 3*21 and the 7-Row with 7*9 in the number “63”.
   In the second case (7 * 9), the 63 would be suitable as a “large number”.
   (It has to be verified if such overlaps of multiple pairs of factors in one number would not increase the cryptographic certainty of a “large number”, due to the fact that the factoring here is less obvious)

    The 12th installment of this work introduced the Tietken-Castell-Prime-Algorithm. It estimates whether a given number is a prime number, an uneven number with the endings 1,3,7 or 9 (no number with the endings 0,2,4,5,6,8) or a so called “large integer“, which results from multiplying two prime numbers. It does so via the position that the numbers assume in a regular, two-stepped (only within the steps from 3 to 7 there is a 4-step in every 10th-row; skipping the number 5) and continuous row of numbers.
   This continuation of this 12th part displays the aforementioned procedure once more shortly in a corrected form, due to the tendency of error when breaking the “registry“ down into partial registries for each respective row of numbers (how it was proceeded in the first part of the 12th part of this work).

   Thus, in this second part of the 12th part there is only one registry for all possible rows of numbers introduced. Repeats, gaps and errors are avoided this way. Furthermore, numbers ending with “-5“, are consequently excluded from the registry.

   This presentation method primarily avoids confusion in the order of the factors. Following, an example: In the presentation of the 3-row, the product 21 does not emerge from 7*3, but from 3*7, because 3 acts as the counting factor in the 3-row, whereas 7 is only the counted factor. In other words: In the row of numbers, 7 appears later than 3 and does not “communicate“ retrograde with it.The initial “partners of communication“ of number 7 are the 7 itself, followed by 9, 11, 13, 17, 21, etc.
   The 3-row therefore shows as: 3, 9 (3*3), 21 (3*7) , 27 (3*9), 33 (3*11), 39 (3*13) etc. The 7-row as: 7, 49 (7*7), 63 (7*9), 77 (7*11), 91 (7*13), 119 (7*17) etc.

   This taken into account, the shown blocks in the 12th (previous) part of this work shouldn’t beginn unified with 1 or 3, but should have started with 3 in the first block, 7 in the second one, and 9 in the third block. The 5-block shouldn’t have been mentioned at all. Later blocks should have started with 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41 etc.
   Multiple assignments, meaning numbers hinting towards multiple factors on the „left side“, are mentioned here in the registry. Although a single pair of factors is sufficient to clarify that the given number is in fact not a prime one, in order to determine that this number is a so called “large integer“, it has to be recognized whether any of the underlying factors were prime numbers. The algorithm saves the as primes numbers identified numbers, so that the aforementioned question is answered at any time.
   Their status of being prime or not of factors used in this registry is not noted, though this information will be necessary for the larger factors.

   Table (called Tietken-Castell-Register):
   Here the initial 280 numbers with respect of the 3-row, 7-row, 9-row, 11-row, 13-row and 17-row:

   1 PRIME 3 (3*3=9) PRIME 7 (7*7=49) (3*3=9) 9 (9*9=81)
   PRIME 11 (11*11=121) PRIME 13 (13*13=169) PRIME 17 (17*17=289) PRIME 19 (19*19=361)
   (3*7=21) 21 (21*21=441) PRIME 23 (23*23=529) (3*9=27) 27 (27*27=729) PRIME 29 (29*29=841)
   PRIME 31 (31*31=961) (3*11=33) 33 (33*33=1089) PRIME 37 (37*37=1369) (3*13=39) 39 (39*39=1521)
   PRIME 41 (41*41=1681) PRIME 43 (43*43=1849) PRIME 47 (47*47=2209) (7*7=49) 49 (49*49=2401)
   (3*17=51) 51 (51*51=2601) PRIME 53 (53*53=2809) (3*19=57) 57 (57*57=3249) PRIME 59 (59*59=3481)
   PRIME 61 (61*61=3721) (3*21=63 und 7*9=63) 63 (63*63=3969) PRIME 67 (67*67=4489) (3*23=69) 69 (69*69=4761)
   PRIME 71 (71*71=5041) PRIME 73 (73*73=5329) (7*11=77) 77 (77*77=5929) PRIME 79 (79*79=6241)
   (3*27=81 und 9*9=81) 81 (81*81=6561) PRIME 83 (83*83=6889) (3*29=87) 87 (87*87=7569) PRIME 89 (89*89=7921)
   (7*13=91) 91 (91*91=8281) (3*31=93) 93 (93*93=8649) PRIME 97(97*97=9409) (3*33=99 und 9*11=99) 99 (99*99=9801)
   PRIME 101 (101*101=10.201) PRIME 103 (103*103=10609) PRIME 107 (107*107=11449) PRIME 109 (109*109=11881)
   (3*37=111) 111 (111*111=12321) PRIME 113 (113*113=12769) (3*39=117 und 9*13=117) 117 (117*117=13689) (7*17=119) 119 (119*119=14161)
   (11*11=121) 121 (121*121=14641) (3*41=123) 123 (123*123=15129) PRIME 127 (127*127=16129) (3*43=129) 129 (129*129=16641)
   PRIME 131 (131*131=17161) (7*19=133) 133 (133*133=17689) PRIME 137 (137*137=18769) PRIME 139 (139*139=19321)
   (3*47=141) 141 (141*141=19881) (11*13=143) 143 (143*143=20449) (3*49=147 und 7*21=147) 147 (147*147=21609) PRIME 149 (149*149=22201)
   PRIME 151 (151*151=22801) (3*51=153 und 9*17=153) 153 (153*153=23409) PRIME 157 (157*157=24649) (3*53=159) 159 (159*159=25281)
   (7*23=161) 161 (161*161=25921) PRIME 163 (163*163=26569) PRIME 167 (167*167=27889) (13*13=169) 169 (169*169=28561)
   (3*57= 171 und 9*19=171) 171 (171*171=29241) PRIME 173 (173*173=29929) (3*59=177) 177 (177*177=31329) PRIME 179 (179*179=32041)
   PRIME 181 (181*181=32761) (3*61=183) 183 (183*183=33489) (11*17=187) 187 (187*187=34969) (3*63=189 und 7*27=189 und 9*21=189) 189 (189*189=35721)
   PRIME 191 (191*191=36481) PRIME 193 (193*193=37249) PRIME 197 (197*197=38809) PRIME 199 (199*199=39601)
   (3*67=201) 201 (201*201=40401) (7*29=203) 203 (203*203=41209) (3*69=207 und 9*23=207) 207 (207*207=42849) (11*19=209) 209 (209*209=43681)
   PRIME 211 (211*211=44521) (3*71=213) 213 (213*213=45369) (7*31=217) 217 (3*73=219) 219 (219*219=47961)
   (13*17=221) 221 (221*221=48841) PRIME 223 (223*223=49729) PRIME 227 (227*227=51529) PRIME 229 (229*229=52441)
   (3*77=231 und 7*33=231 und 11*21=231) 231 (231*231=53361) PRIME 233 (233*233=54289) (3*79=237) 237 (237*237=56169) PRIME 239 (239*239=57121)
   PRIME 241 (241*241=58081) (3*81=243 und 9*27=243) 243 (243*243=59049) (13*19=247) 247 (247*247=61009) (3*83=249) 249 (249*249=62001)
   PRIME 251 (251*251=63001) (11*23=253) 253 (253*253=64009) PRIME 257 (257*257=66049) (7*37=259) 259 (259*259=67081)
   (3*87=261 und 9*29=261) 261 (261*261=68121) PRIME 263 (263*263=69169) (3*89=267) 267 (267*267=71289) PRIME 269 (269*269=72361)
   PRIME 271 (271*271=73441) (3*91=273 und 7*39=273) 273 (273*273=74529) PRIME 277 (277*277=76729) (3*93=279 und 9*31=279) 279 (279*279=77841)
   PRIME 281 (281*281=78961) PRIME 283 (283*283=80089) (7*41=287) 287 (287*287=82369) (17*17=289) 289 (289*289=83521)
   (3*97=291) 291 (291*291=84681) PRIME 293 (293*293=85849)

   ……and further up to infinity!

   A comment concerning the table:
   What is written on the left hand side of the numbers in the registry is of importance.
   These are the two or more factors in the case of a non-prime number, which have “produced“ the respectively investigated number.
   The multiplications written on the right hand side display the future products, which emerge when newly appearing number is multiplied by itself.
   Due to the fact that every new number which founds its own new row of numbers, has to start their multiplications on the right hand side with the smallest possible number, which is, in fact, itself.

   B)
   Further possibilities of prime number detection:

   At the end of this work we additionally introduce how The Tietken-Castell-Prime-Algorithm detects prime numbers also in a direct way, but not only by their appearance and location in the registry.
   „Direct“ means, that therefore no registry (as the one above) is necessary. Every single given number can be investigated with use of the algorithms techniques, whether it is prime number or something different.
   When the square root of a number with 1,3,7 or 9 does not deliver a smooth result (but a number with decimal digits), then it is clear that: (a) This number is not made up of two similar factors and (b) a prime number could be evident. Because prime numbers do not contain square roots.
   These uneven numbers with the end-digits 1,3,7 or 9 could c) also be a product of two different prime numbers.
   In order to clarify, whether this number is a prime one or a product of different factors, this number has to be divided by each number of the always recurring and constantly equal row of numbers, meaning by 3, 7, 9, 11, 13, 17, 19, 21 etc.
   The amount of divisions and and used divisors is determined by the location of the dividend within the Tietken-Castell Registry.

   As an example, the number 221 is located in the 22. row at the first place, following 88 calculations are necessary (22 rows x 4 numbers). Subtracted the number 1 in the first row, as well as the 3, 7 and 9 in the 22nd row, we are left with 84 divisions for the number 221, which decide from what factors the (non-prime!) 221 is made up.
   After finding these factors, it is additionally detectable if these are prime, therefore if 221 is a so called “large integer“.

   However, according to probability, these 84 calculations will shrink down to on average 42 calculations in applied praxis. In the concrete case of the here randomly taken 221, only 5 actual divisions (by 3, 7, 9, 11, 13) are necessary, in order to obtain the two factors 13 and 17 via the divisor 13.
   The latter are two unequal factors, which as well are prime, thus the 221 is in fact a large integer, which can be used for the RSA-Encryption.
   If there is any dought, if a larger prime number is really a prime number, the above mentioned procedure can be analogously repeated.
   With the detection (registry or via division) of these two prime factors 13 and 17, the necessity of a factorization of 221 and every other investigated number is omitted. Practically, both shown procedures are able to omit a factorization of “large integers“ (given an adequately large registry or amount of divisions).

   In order to shorten the procedure of multiple divisions, an additional service of the Tietken-Castell-Prime-Algorithm is presented: The algorithm is able to identify which factors took part in a multiplication for every number, no matter how many digits are present, by looking at the ending digits.
   So (if factors that end with a 1 are excluded), a 1 for the supposed product hints towards possible factors with the last digits 3 and 7 or 9 and 9,
   a 3 hints towards factors with the last digits 7 and 9,
   a 7 hints towards factors with the last digits 3 and
   a 9 towards 3 and 3 or 7 and 7.
   With this additional step the maximally possible amount of divisions for every number could be, depending on the respective ending digit, drastically reduced, at least splited into thirds.
   The aforementioned paragraph described how the interpretation of numbers is possible without the assistance of the the Tietken-Castell-Registry, provided that the procedure of the Tietken-Castell-Prime-Algorithm is consequently applied.

   C)
   The direct solution without a registry and its methods via the novel, not yet introduced Castell-Fact-Algorithm:

   Without the preparatory work of having created a large register yourself or being able to access another register, or if the large number to be broken down has too many digits, the Castell-Fact-Algorithm has to be utilized, which (without brute force) is able to quickly and inexpensively break down unlimited numbers into their prime numbers.

   Nikolaus Graf zu Castell-Castell
   Dipl. Vw. (University Hamburg)
   　　

   Prague Research Institute
   Zug (CH) und Prague (CR)
   mob. 00420 778 037 633
   fix line 00420 226 223 026

   https://zenodo.org/record/4080970#.X4XAoHVCSws

   https://zenodo.org/record/4076800#.X4XBAXVCSws

8. #8 Cosma Rules

   7. November 2020

   Wegen faktischer Vorhersehungskraft eines menschlichen Bewusstseins müssen neben gamma[n](t) noch precog[n](psi) existieren, welches dem immateriellen Bewusstsein zeitunabhängige Sprünge in die nähere Zukunft erlaubt, um die Gegenwart so zu ändern, dass eine andere Zukunft eingeleitet wird.

   Berichtet wurde über eine schwangere Amerikanerin aus Chicago, die ihren Fruchtwasserembolie-Tod vorhersah und die Ärzte plötzlich aufforderte, unübliche Medizintechnik während der Geburt bereitzuhalten.
   Sie starb, war 37 Sekunden klinisch tot, konnte dank der unüblichen Medizintechnik innerhalb einer Minute zurückgeholt werden, bevor ihr Gehirn zu Broccoli wurde.

   Ein precog[](psi) windet sich spiralig um ein gamma[](t) und ist phasenverschoben, weswegen Thilo sie im Bild Homotopy_curves nicht abbilden kann.

   Hybris ist, dass jüdische Mathematiker, die an den jüdischen Gott Jahwe glauben,
   (The Assyrian Tree of Life: Tracing the Origins of Jewish Monotheism and Greek Philosophy)
   ihrem Gott mathematisch verbieten, precog[](psi) zu erschaffen, mit deren Hilfe er alles sehen kann (Bette Midler: God is watching us from a distance).

   Mathematik

   0-dimenionale t-Zahlen werden in gamma(t) von x nach y schrittweise größer

   Unterdrückte Physik

   Eine komplexe t-abhängige Wellenfunktion eines Standard-Modell-Partikels hat eine minimale Amplitude (10^-40? Meter) und gamma(t) muss ein Schlauch sein, um in sich eine Wellenfunktion und ihre Amplitude zu ermöglichen.

9. #9 Verfassungsschelm

   9. November 2020

   Der deutsche YOTA-Skandal

   Ferdinand Springer machte den Springer Verlag, der Verlag der jüdischen Familie Springer, nach dem ersten Weltkrieg über Deutschland hinaus zum führenden Verlag für mathematische Literatur.

   Später wurden auch Physik-Lehrbücher vom Verlag veröffentlich.
   Sind im Springer Verlag alternative Beschreibungen der Physik und der Kosmologie erlaubt, in der die Zahl 0 und das Symbol Unendlich aus realen physikalischen Gründen nicht erlaubt sind?
   Nein. Der Springer Verlag verbietet zensierend das Erwähnen einer Abbruch-Schranke YOTA, die das reale unvollkommene Physikreich vom virtuellen idealen Mathematiksimulator trennt, weswegen Physiklehrbuchautoren eine mathematische Krücke diktatorisch aufgezwungen wird z.B. delta(x), um über den mathematischen Null-Radius eines mathematischen Elektrons integrieren zu können,

   Gibt es physikalische YOTA-Schranken in deutschen Physiklehrbüchern oder in deutschen Physik-Vorlesungen und -Übungen?
   – Nein, sie zensieren alle, und so reicht das elektrische Feld des Kerns eines ionisierten Atoms in jeder Physik-Vorlesung bis in die mathematische Unendlichkeit, obwohl Rydberg-Experimente bewiesen, das ab einer mikroskopischen Schranke (10^-8 Meter) das elektrische Feld des ionisierten Kerns nicht mehr wirkt und kein freies Elektron jenseits dieser Schranke einfangen kann.
   – Auch in der Gravitations-Physik gibt es eine Abbruch-Schranke, weswegen Albert Einstein Trägheit nicht als träge gravitierende Wechselwirkung einer lokalen Masse mit allen gravitierenden Masse-Teilchen des Universums mathematisch darstellte, welches aus stabilen Existenz-Gründen nicht unendlich viel Masse beherbergen kann, sondern mit einem lapidaren Äquivalenz-Prinzip träge Masse und schwere Masse gleichsetzte.
   – Wenn das Universum eine kompakte abgeschlossene Menge bzw. physikalisch beschränkt ist, müssen physikalische elektrische Feldlinien wegen des Energieerhaltungssatzes und ihrer Lichtgeschwindigkeit am kompakten Rand nach n Milliarden Jahren reflektiert werden, was in Physik-Vorlesungen nicht erwähnt wird, wenn es sich in Wahrheit um mathematische Mystik-Vorlesungen handelt.
   – Ein mit halber Lichtgeschwindigkeit um die Symmetrie-Achse rotierender, materieller, physikalischer Tipler-Zylinder benötigt nicht unendlich viel Masse, um eine Singularität zu erzeugen, weil die beschränkte Physik nicht bis unendlich zählen kann, sondern bereits ab einer physikalischen Abbruch-Schranke eine Singularität erzeugt.
   – Und so weiter …

   Ein immatrikulierter Physik-Student, Mitglied einer unabhängigen Partei, kann einen unabhängigen Partei-Rechtsanwalt bitten, seiner deutschen Physik-Fakultät eine Unterlassungserklärung zuzustellen, die verlangt, dass in den Physik-Vorlesungen und -Übungen die Physik-Schranken grundsätzlich nicht verletzt werden bzw. das Streben von Prozessen gegen 0 (Elektronen-Radius) oder Unendlich (Elektrisches Feld, rotierender Tipler-Zylinder) als gefährlicher, gesetzeswidriger, das Allgemeinwohl gefährdender Physik-Kollapsar zu kennzeichnen ist.
   In der Unterlassungserklärung wird darauf hingewiesen, dass jeder Physik-Student dieser Mystik-Fakultät arglistig getäuscht und in seiner freien Berufswahl verfassungswidrig eingeschränkt wird, weil er in dieser Mystik-Anstalt zum talmudischen sorglosen Mathematiker anstatt zum verantwortungsbewussten Physiker ausgebildet wird.

   Wird die auf der deutschen Verfassung beruhende Unterlassungserklärung, hinter der ein unmittelbar und gegenwärtig betroffener Physik-Student einer Physik-Fakultät steht, von einem talmudisch geprägten Gericht trotz Gehörsrüge ungehört/unsachverständig abgelehnt, kann mit einer Verfassungsbeschwerde durch den Subsidiaritätsbruch zum Bundesverfassungsgericht zeitlos schnell gesprungen und die dazwischen liegenden Instanzen des Rechtswegs in Nullzeit übersprungen werden.

   Sollte das Bundesverfassungsgericht eine solche Beschwerde wie üblich unbegründet oder ähnlich wie im LHC-Urteil unsachverständig abweisen, dann weiß Thilo, dass er die DDR nicht verlassen hat:

   Wenn die Gaußsche Krümmung aus Geldgründen physikalisch gebrochen werden kann und sie somit keine fundamentale Invariante für die Differentialgeometrie der Flächen im 3-dimensionalen Physik-Raum ist, dann ist sie zu brechen.

   Übrigens, jeder Richter ist talmudisch befangen, der Physikraum mit Hilfe von mathematischen Invarianten Unzerstörbarkeit andichtet.
   Somit „muss“ mit einem Befangenheitsantrag festgestellt werden, ob ein Richter an den zerbrechlichen Physikraum oder den unzerstörbaren Mathesimulator glaubt.
   Wenn eine Kammer des Bundesverfassungsgerichts an den unzerstörbaren Mathesimulator glaubt, bedeutet das nicht, dass alle anderen Kammern bzw. Richter das auch tun, was wiederum bedeutet, dass die Beschwerde solange eingereicht werden muss, bis alle Richter ihren Glauben offenbarten.
   #mimimibe

10. #10 Thilo

    9. November 2020

    Sorry, ich hab’ den Witz nicht verstanden.

11. #11 rolak

    9. November 2020

    den Witz

    Den Witz, Thilo? Welchen Witz? Das ist Voller Ernst, wenn nicht gar Schall und Rauch.
     
    Hat mir der da verraten.

12. #12 Verfassungsschelm

    9. November 2020

    Wegen (kurzer) Verjährungsfristen kann eine Verfassungsbeschwerde wegen überhörter Gehörsrüge nicht beliebig oft eingereicht werden.
    Weil eine Kammer (3 Richter) eines Senats des Bundesverfassungsgericht für die 8 Richter seines Senats sprechen und urteilen darf, kann man sich den Befangenheitsantrag, auf den wegen grundgesetzlichen Anspruchs auf rechtliches Gehör auch geantwortet werden muss, gegen alle 16 Richter sparen und innerhalb der (kurzen) Verjährungsfrist die Verfassungsbeschwerde an den 1ten Senat und die gleiche Verfassungsbeschwerde an den 2ten Senat schicken.
    Werden die 2 Verfassungsbeschwerden jeweils von einer Kammer eines Senats abgewiesen, ist bewiesen worden, dass 16 talmudische Bundesverfassungsrichter unsachverständig an den Mathesimulator und seine unzerstörbaren Invarianten glauben müssen und jedes Physik-Experiment ihrer Regierung bewilligen müssen, was im Sinne Gödels ein schöner Beweis für die Existenz einer Wissens-Diktatur ist.

    Werden 2 identische, gut begründete Verfassungsbeschwerden für 2 Senate, die alle Pflichten des Merkblatts des Bundesverfassungsgerichts erfüllen, von der gleichen Kammer eines Senats unbegründet abgewiesen, erhält man einen DDR-Witz.

    Nicht witzig ist, dass Thilo
    – (und Millionen andere) von einer demokratischen Regierung über neue demokratische BRD-Grundrechte nach dem Fall der DDR nicht aufgeklärt wurde, und er nicht weiß, wie ein unrechtstaatlicher Dezernatswechsel §101 und §103 des Grundgesetzes verletzen kann, was wie eine überhörte Gehörsrüge im Fall eines überhörten Verjährungseinwands oder im Fall unnötiger Kosten eines Kostenfestsetzungsbescheids den zeitlosen Sprung durch den Subsidiaritätsbruch nach Karlsruhe ermöglicht.
    – und jeder andere Mathematiker/Physiker trotz Hinweis auf den Bruch in einer Wissens-Diktatur kein mathematisches Kontinuum erschaffen darf, das zeitlose Raumdurchquerung oder Präkognition ermöglicht.
    – sich für Gödels Kritik an der amerikanischen Verfassung interessiert aber seine deutsche Verfassung und dessen Interpretation durch das Bundesverfassungsgericht nicht studiert, um demokratische Diktatur zu entdecken, die ihn weder über seine demokratischen Grundrechte noch über mögliche Verletzungen seiner Grundrechte durch nicht demokratische Richter aufklärte.

13. #13 Quanteder

    10. November 2020

    „ Die Konstruktortheorie des Lebens zeigt also ausdrücklich, dass die natürliche Selektion nicht die Existenz eines Anfangsrezepts voraussetzen muss, das Wissen enthält, um loszulegen. Es zeigt, dass alle Rezepte, die wir in Lebewesen finden, keine Ad-hoc-, biozentrischen oder mysteriösen Gesetze der Physik erfordern, um aus elementaren Anfangskomponenten zu entstehen. Sie brauchen nur die Gesetze der Physik, um die Existenz digitaler Informationen sowie ausreichend Zeit und Energie zu ermöglichen, die nicht spezifisch für das Leben sind. Dies fügt einen weiteren tiefen Grund hinzu, warum eine Vereinheitlichung unseres Verständnisses der Phänomene des Lebens und der Physik möglich ist. Was auch immer die Gesetze der Physik uns nicht verbieten, wir können es tun. Ob wir es tun oder nicht, hängt davon ab, wie viel Wissen wir schaffen. Es liegt an uns.“

    —> Maschinenübersetzung letzter Absatz aus https://aeon.co/essays/how-constructor-theory-solves-the-riddle-of-life

    Hier sehe ich Lösungsangebote für die in #7, #8, #9 und auch #10 aufgeworfenen Fragen . . . .. 🙂

14. #14 Irish

    France, Hazebrouck

    12. Februar 2021

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15. #15 Theorema Magnum – Mathlog

    15. April 2021

    […] linearer Gleichungssysteme Das Simplex-Verfahren Das WKS-Abtasttheorem Adelische Poisson-Summation Der Vergleichssatz von Rauch Die Berechnung des Kobordismusrings Die Endlichkeit der Homotopiegruppen von Sphären Der […]

16. #16 Nikolaus Graf zu Castell-Castell

    Prague

    28. Juni 2021

    RSA, PGP, Bitcoin & Co.
    20 Teil-Überlegungen, ausschliesslich mit Multiplikation und Division ein zusammengesetztes Produkt in seine Primfaktoren zu zerlegen.
    Ausgangs-Ueberlegung bis Ueberlegung 12.

    https://zenodo.org/record/5032371#.YNib_kBxews Ausgangs-Ueberlegung
    https://zenodo.org/record/5032333#.YNiZb0Bxews Ueberlegung 1
    https://zenodo.org/record/5032509#.YNiY-UBxews Ueberlegung 2
    https://zenodo.org/record/5032944#.YNiaakBxews Ueberlegung 3
    https://zenodo.org/record/5033157#.YNia2EBxews Ueberlegung 4
    https://zenodo.org/record/5033196#.YNibPEBxews Ueberlegung 5
    https://zenodo.org/record/5034688#.YNicb0Bxews Ueberlegung 6
    https://zenodo.org/record/5034337#.YNiZxEBxews Ueberlegung 7
    https://zenodo.org/record/5034361#.YNiXg0Bxews Ueberlegung 8
    https://zenodo.org/record/5034678#.YNibj0Bxews Ueberlegung 9
    https://zenodo.org/record/5034447#.YNiem0Bxews Ueberlegung 10
    https://zenodo.org/record/5034485#.YNieAkBxews Ueberlegung 11
    https://zenodo.org/record/5034630#.YNic4UBxews Ueberlegung 12

    Fortsetzung folgt