l - OPUS at UTS - University of Technology Sydney
... Here, we produce similar results involving the arctan function and terms from the sequences {£/„} and {Vn}. Some of our results are equivalent to those obtained in [13] but most are new. We also obtain results involving the arctanh function, all of which we believe are new. ...
... Here, we produce similar results involving the arctan function and terms from the sequences {£/„} and {Vn}. Some of our results are equivalent to those obtained in [13] but most are new. We also obtain results involving the arctanh function, all of which we believe are new. ...
Lecture 6: RSA
... and we can combine the two pairs in any way we want. In particular, if N = 15 then the four solutions are 1, 4, 11 and 14. For example 4 = 1 (mod 3) and 4 = −1 (mod 5). For one of the interesting solutions (i.e. not ±1) it turns out that GCD(N, x − 1) gives a nontrivial factor of N. One can prove th ...
... and we can combine the two pairs in any way we want. In particular, if N = 15 then the four solutions are 1, 4, 11 and 14. For example 4 = 1 (mod 3) and 4 = −1 (mod 5). For one of the interesting solutions (i.e. not ±1) it turns out that GCD(N, x − 1) gives a nontrivial factor of N. One can prove th ...
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.