a 1 - Bowie High School
... terms of the sequence as the range. Values in the domain are called term numbers and are represented by n. Instead of function notation, such as a(n), sequence values are written by using subscripts. The first term is a1, the second term is a2, and the nth term is an. Because a sequence is a functio ...
... terms of the sequence as the range. Values in the domain are called term numbers and are represented by n. Instead of function notation, such as a(n), sequence values are written by using subscripts. The first term is a1, the second term is a2, and the nth term is an. Because a sequence is a functio ...
Number Theory and Combinatorics
... table such that for each i, 1 ≤ i ≤ 9 the product of the numbers in row i equals the product of the numbers in column i. Solution The key observation is the following: if row k contains a prime number p > 40, then the same number must be contained by column k, as well. Therefore, all prime numbers f ...
... table such that for each i, 1 ≤ i ≤ 9 the product of the numbers in row i equals the product of the numbers in column i. Solution The key observation is the following: if row k contains a prime number p > 40, then the same number must be contained by column k, as well. Therefore, all prime numbers f ...
REPRESENTATIONS OF INTEGERS BY QUADRATIC FORMS As
... d2 4a (8k + 7) = (2b )2 (d′2 )4a (8k + 7) = 4a+b (8k ′ + 7). In other words, to show that no integer of the form 4a (8k + 7) is a sum of 3 rational squares, it suffices to show that no integer of the form 4a (8k + 7) is a sum of three integral squares. So let us now show this. Step 1: We observe that ...
... d2 4a (8k + 7) = (2b )2 (d′2 )4a (8k + 7) = 4a+b (8k ′ + 7). In other words, to show that no integer of the form 4a (8k + 7) is a sum of 3 rational squares, it suffices to show that no integer of the form 4a (8k + 7) is a sum of three integral squares. So let us now show this. Step 1: We observe that ...
Mathematics Course 111: Algebra I Part I: Algebraic Structures, Sets
... In order to prove that certain properties hold for all positive integers, the Principle of Mathematical Induction is often employed. This principle is a consequence of the following important property of subsets of the set N of positive integers: The Principle of Mathematical Induction (for subsets ...
... In order to prove that certain properties hold for all positive integers, the Principle of Mathematical Induction is often employed. This principle is a consequence of the following important property of subsets of the set N of positive integers: The Principle of Mathematical Induction (for subsets ...
Chapter 6 Integers and Rational Numbers
... proof to see that we didn’t cancel 0. How do we find q and r in practice? Use the long division algorithm that you learned at school. The way it works is very similar to the proof we gave above. To divide a by b, you find (by trial division) the largest q such that bq ≤ a, and then put r = a − bq. O ...
... proof to see that we didn’t cancel 0. How do we find q and r in practice? Use the long division algorithm that you learned at school. The way it works is very similar to the proof we gave above. To divide a by b, you find (by trial division) the largest q such that bq ≤ a, and then put r = a − bq. O ...
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.