Modular Arithmetic
... We want to show that for every m there exists an n such that n + 1, n + 2, . . . , n + m are not prime. Let us assume that n + 1 is divisible by 2 and that n > 2. Then n + 1 is not a prime number. Now n + 2 is not divisible by 2. However, we could assume that n + 2 is divisible by 3 and n + 2 > 3. ...
... We want to show that for every m there exists an n such that n + 1, n + 2, . . . , n + m are not prime. Let us assume that n + 1 is divisible by 2 and that n > 2. Then n + 1 is not a prime number. Now n + 2 is not divisible by 2. However, we could assume that n + 2 is divisible by 3 and n + 2 > 3. ...
Integers Review (+, -, x, div ) - middle-school
... • Share with your partner the rules for Adding and Subtracting Integers. (1’s start) • Share with your partner the rules for Multiplying and Dividing Integers. (2’s start) ...
... • Share with your partner the rules for Adding and Subtracting Integers. (1’s start) • Share with your partner the rules for Multiplying and Dividing Integers. (2’s start) ...
![Maximum subsets of (0,1] with no solutions to x](http://s1.studyres.com/store/data/004884619_1-aef9649f88c5cc5f6ca0274124904717-300x300.png)
Full text
... digit length n one of two things must happen; either in the sequence of iterates we find one with fewer than n digits, which property will then persist, or else the sequence becomes periodic eventually with all the numbers in the period having n digits. Within a period, the period-length p, is the n ...
... digit length n one of two things must happen; either in the sequence of iterates we find one with fewer than n digits, which property will then persist, or else the sequence becomes periodic eventually with all the numbers in the period having n digits. Within a period, the period-length p, is the n ...
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.