Proof isn't always possible

 Newton's method is known to converge slowly, and I do not have an associated margin of error at this time. Newton proved: x(k+1) = x(k) - [x(k)^n + C]/[n * x(k)^n-1] converges on the nth root of C. My first step is to rewrite x(k) as (a/b)(k) or a(k)/b(k) From this we obtain: (a/b)(k+1) = (a/b)(k) - [(a/b)(k)^n - C / n * (a/b)(k) ^ (n-1)] Focusing on x(k), we begin: I do not know how to use the editor that does notation.

Six sided dice can result in random numbers, when the face value is converted to Base 10, DESPITE displaying no "0." I suspect that Number Theory, as a class, would have taught me more than some finite number of Arithmetic tricks. It is my understanding that Number Theory investigates how these apparent coincidences are true using Algebra. It has been brought to my attention that, IF Erasthmus's Sieve is correct, THEN the modified Sieve in the previous post is also correct, as follows: Erasthmus used a positive integer number line, and stepped by the relevant n. In the sieve I uncovered (seeking a way to improve compactness in computer RAM) I start with odd integers, as a number line, and as a practical matter strike by 2*n. Explicitly, if we start with 3 and strike every third value, for example, we step over an even number in between each odd number skipped, resulting (algebraically) in striking by 2*3, or 6. Under this constraint, 3 being odd, we strike all ODD multip

 I wrote before, asking if the modification on Erasthmus's Sieve I discovered was still Erasthmus's Sieve. I am convinced it is not the Sieve of Erastosthenes, since the odd numberline is not a geometric progression. I allude to it as a Modified Erasthmus's Sieve, while open to correction. Earlier I demonstrated n < 100. Although it took me a long time to present the current entry, I did understand that n < 100 was not sufficient. Below I present a clearer picture of how it develops, using n<=361, or 19^2. I don't know enough Number Theory to offer Proof. In the first slide, I enumerate 2 literal, and develop the odd number line up to 361. I then start with 3 as the next unstruck number, and blindly strike by 3s. The result leaves NO MULTIPLES OF 3 UNSTRUCK. In the second slide, I start with the next unstruck number (5) and blindly strike by that number. The result leaves NO MULTIPLES OF 5 UNSTRUCK. By observation of experiment, multiples of 3 x 5 (15) are doub

Retraction: It is with regret that I do not delete the previous post. I am an individual, not an organization, and I failed to adequately double-check my results, but I am leaving the post to maintain some accountability. I am able to definitively say that the script for 2chars_from_file.py DOES NOT yield an exhaustive list. It develops the ciphertext for all 2 Char combinations of the FIRST 10, three character ciphertexts. This is not due to a logic error, but rather there is a virus that corrupts the indentation of the two write statements.  This may be verified by downloading the relevant code, and examining it, and running it if needed, against the relevant datafile. It is difficult to be exhaustively correct about ALL of the scripts. This calls ALL of my results in question, with regard to the qualification of a mathematical group. If Vigenere IS a group, the resulting substitution is still valid, but there exist 6 Char passwords that will accomplish the same encryption in ONE STE

I have for some time speculated that the 1500's Cipher, " Vigenere ," is not a mathematical  group . Not to bury the lead, IT IS NOT!  However, this is not completely satisfying for the question I was asking.   Vigenere could be represented as a Matrix function such that f(P) = C and f(C) = P. (P for plaintext, C for ciphertext.) Early, I observed that when the operation is repeated with passwords of differing lengths, the periodicity is equal to the product of the unique prime factors of the password lengths. In encryption, there is also a concept of the amount of ciphertext necessary to calculate or infer the password. This is called the " Unicity distance ." In Vigenere, the unicity distance is the same as the period. Since the idea in encryption is to extend the period, I amused myself with double and triple encrypting. Around 1995, I worried that a double encrypted message of a specific period might be comparable to a message encrypted ONCE, with a single p

I speculated in my last article that using cryptocurrency in gaming would improve the gaming experience by making allocation of computer resources strategic. By application, each unit of crypto would be "backed," by the plane or tank purchased with it. It is worth speculating about the following complicating factors:  As you mine, and (for example) pay soldiers, you accumulate more currency than materiel. An available decision to add, is whether or not, or possibly at what time to " fork " the cryptocurrency, and stop accepting new coins. Soldier's pay might be lost to the game, if there is no economy for the soldiers to expend it on. If they buy food, it is perishable, but some currency might be preserved by building restaurants, or having service industry staff buy televisions, phones, and stereos. Meanwhile, Soldiers might pay for uniforms, and boots, if they are not provided by their "government."  If I were designing this, I could contrive Rock co