Application of Linear Sequences to Cryptography
... In the movie Sherlock Holmes: A Game of Shadows, Professor Moriarty attempts, and at first succeeds, in deceiving Sherlock Holmes about where he will strike his next target. Rather than being obvious by whispering to his evil henchmen the location of the next assassination, he chooses to put the loc ...
... In the movie Sherlock Holmes: A Game of Shadows, Professor Moriarty attempts, and at first succeeds, in deceiving Sherlock Holmes about where he will strike his next target. Rather than being obvious by whispering to his evil henchmen the location of the next assassination, he chooses to put the loc ...
Notes on Algebraic Numbers
... This result is useful as it shows that some algebraic numbers aren’t algebraic integers. The most important result in this section is that sums and product of algebraic numbers (resp. algebraic integers) are also algebraic numbers (resp. integers). Before we can prove this we need an alternative des ...
... This result is useful as it shows that some algebraic numbers aren’t algebraic integers. The most important result in this section is that sums and product of algebraic numbers (resp. algebraic integers) are also algebraic numbers (resp. integers). Before we can prove this we need an alternative des ...
Slides
... Theorem: The only way to make 2i in i stages is by repeated doubling Assumption: Let 2i+1 be the first counter-example to the theorem. To make 2i+1 at stage i+1 requires that some number of size at least 2i be present at stage i. By previous result such a number could not be larger than 2i, and ...
... Theorem: The only way to make 2i in i stages is by repeated doubling Assumption: Let 2i+1 be the first counter-example to the theorem. To make 2i+1 at stage i+1 requires that some number of size at least 2i be present at stage i. By previous result such a number could not be larger than 2i, and ...
Chapter Summary and Summary Exercises
... exercises carefully. References are provided to the chapter and section for each exercise. If you have difficulty with any exercises, go back and review the related material. ...
... exercises carefully. References are provided to the chapter and section for each exercise. If you have difficulty with any exercises, go back and review the related material. ...
Collatz conjecture
The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.Take any natural number n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called ""Half Or Triple Plus One"", or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called oneness.Paul Erdős said about the Collatz conjecture: ""Mathematics may not be ready for such problems."" He also offered $500 for its solution.