Chapter 2 – Integers
... In this chapter we will be using a new set of numbers. Some of you may never have seen this set of numbers and for others it may seem familiar but scary. The new set of numbers is called the integers. Integers are the whole numbers and their opposites. An opposite is a number on the number line that ...
... In this chapter we will be using a new set of numbers. Some of you may never have seen this set of numbers and for others it may seem familiar but scary. The new set of numbers is called the integers. Integers are the whole numbers and their opposites. An opposite is a number on the number line that ...
Math 259: Introduction to Analytic Number Theory Elementary
... this argument because I was able to adapt it as the punchline of my doctoral thesis; see [Elkies 1987].] The case of χ(p) = +1 is only a bit trickier.2 For instance, to prove Dirichlet for (q, a) = (4, 1), let p1 = 5 and Nn = 4Pn2 + 1, and invoke Fermat’s theorem on the prime factors of x2 + y 2 . A ...
... this argument because I was able to adapt it as the punchline of my doctoral thesis; see [Elkies 1987].] The case of χ(p) = +1 is only a bit trickier.2 For instance, to prove Dirichlet for (q, a) = (4, 1), let p1 = 5 and Nn = 4Pn2 + 1, and invoke Fermat’s theorem on the prime factors of x2 + y 2 . A ...
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.