For example, 2789 is not the answer, because while 89 is prime, 8, 9, 27, 78, 278, and 789 are not prime.
Challenge: Try to solve this in your head without using pencil or paper, or googling if a number is prime or not. Hint: >!The answer is three digits long!<
In https://www.crownsterling.io/2019/09/crown-sterling-decrypts-rsa-asymmetric-public-keys-in-live-demonstration/ the infamous Crown Sterling reports they were able to factor 256-bit RSA keys in just 50 seconds (demo here). Luckily it was that fast, as slowing down the cado-nfs program they plagiarized as their own could have wasted more of humanity's time.
In the same announcement:
>Crown Sterling also announced the completion of a new white paper to be submitted for peer review, βNovel Geometric Methods for Semiprime Factorization,β co-authored by Robert Edward Grant, Co-Founder and CEO, Crown Sterling, and Talal Ghannam PhD., Physicist and Number Theorist.Β The paper includes four different geometric and arithmetic methods for public key (semiprime) factorization and one of the methods titled, βThe Reciprocal Factoring Methodβ includes an analysis of reciprocal values of public keys and their embedded private keys (prime factors) found within their period decimal extensions.
To illustrate this "factoring" method, Grant posted not one, but two images into his Instagram. Here are the captions for the images:
>Identifying Prime Factors from the Reciprocal value of x (BiPrime). Note that in this case the prime factors are 23 and 47.....reviewing the 1/x the first prime factor (47) appears after 23 digits in the decimal extension and the other factor is 23. We have noted several such correspondences that relate to the Periodicity (Repetition Cycle) within the repeating rational of the (1/x)reciprocal.
>
>None of us would have believed that the factors of BiPrimes could be found within the βtrivialβ reciprocity of x. This is beyond astounding to our mathematics and physics team and implies greater meaning to all the Wave-based numbers that form the βabsorptionβ of each βreflectedβ number we consciously perceive: βEvery coin has therefore TWO sidesβ. Now, for the next logical question/s: What can the other numbers in the reciprocal string tell us about the nature of x? Are geometries associated with this seemingly random list of numbers behind the decimal ext
This may seem like a stupid question, but would a decimal expansion of say, 0.00351 be a decimal expansion of only odd numbers or would some strange property of zero negate that?
Look at the following fraction:
3/8
If you divide it out it equals 0.375 in base 10, which is approximately ~0.38.
3/8 = ~.38
The numerator and denominator are the same digits (almost -- with rounding). Is there a fraction where the decimal expression is the same sequence as the numerator and denominator concatenated together?
For example, you can get a little closer with 31/97 = 0.31958... but it's still not exact! How close can you get? Is there a closest number? Or are there unlimited numbers constantly converging?
1/3=0.333333 goes on till infinity but itβs smaller than 0.34
I am just curious if there is a way to easily go from a decimal expansion of some number (say we know that it is rational) to the fraction. For example, given the first hundred digits of a particular fraction (0.0000335328780957335818405539497775522447253952506972554843244935393507210404327412348008887883769028), how could I tell that that is 301/8976265?
Everyone thinks pi is almost certainly normal, but that's a very hard thing to prove. I want to see a proof of this much easier fact, one that doesn't rely on actually computing pi.
Okay, so I should clarify for you nit-picky mathematicians what I mean by complicated. Consider that we take the number of digits needed to represent each value in the sequence 1/n such that n is a natural number in decimal notation. Does this sequence follow some pattern? If so, what causes the "spikes", like at n=7?
{1/2, 1/3, 1/4, 1/5, 1/6, 1/7, ....} <------ The sequence of fractions
{0.5, 0.333, 0.25, 0.2, 0.1666, 0.142857, ...} <------- The sequence of (rough) decimal representations
{1, 1, 2, 1, 2, 6, .... } <-------- The sequence of the number of distinct symbols needed to write each decimal rep. in the above sequence.
I am working on Project Euler Problem #26, and I need to wrap my head around the mathematical concept behind this problem before I attempt to write the code to solve it.
Instead of for going for the brute-force solution (getting the decimal expansion for every number [;1/n;] for [;1 < n < 1000;], then finding a repeating sequence in each number's decimal expansion and then figuring out which one if the largest), I set out to find something within math that would help me find the underlying pattern.
So, after a bit of searching I found that multiplicative order should help me with this specific problem.
My problem is that I don't understand what is being said in that article. The article states:
> The multiplicative order of 10 mod an integer n relatively prime to 10 gives the period of the decimal expansion of the reciprocal of n For example, the haupt-exponent of 10 (mod 13) is 6, and
> 1/(13)=0.076923 repeating
> which has period 6.
I don't understand this. To me, it sounds like the statement is saying 10 % 13 = 6, which is obviously not true.
What exactly does it mean?
title says it all
