Counting
... If N objects are placed into k boxes, then there is at least one box containing at least N/k objects ...
... If N objects are placed into k boxes, then there is at least one box containing at least N/k objects ...
1 - University of Kent
... of the iterates belong to the ring Z[x±1 0 , . . . , xN −1 ]; as a consequence, if all the initial values are 1 (or ±1), then each term of the sequence is an integer. Such sequences were popularized by Gale [8, 9], and subsequently Fomin and Zelevinsky found a useful technique - the Caterpillar Lemm ...
... of the iterates belong to the ring Z[x±1 0 , . . . , xN −1 ]; as a consequence, if all the initial values are 1 (or ±1), then each term of the sequence is an integer. Such sequences were popularized by Gale [8, 9], and subsequently Fomin and Zelevinsky found a useful technique - the Caterpillar Lemm ...
Acta Mathematica et Informatica Universitatis Ostraviensis - DML-CZ
... Diophantus studied the following problem: Find four (positive rational) numbers such that the product of any two of them increased by 1 is a perfect square. He obtained the following solution: -—, ff, T ' W ( s e e -7-)* T h e ^ r s t s e t ° ^ o u r positive integers with the above property was fou ...
... Diophantus studied the following problem: Find four (positive rational) numbers such that the product of any two of them increased by 1 is a perfect square. He obtained the following solution: -—, ff, T ' W ( s e e -7-)* T h e ^ r s t s e t ° ^ o u r positive integers with the above property was fou ...
![[Part 1]](http://s1.studyres.com/store/data/008795717_1-da61206028950a8b76c72065c95ca070-300x300.png)
Introduction to Proofs
... Direct proofs lead from the hypothesis of a theorem to the conclusion. They begin with the premises; continue with a sequence of deductions, and ends with the conclusion. Direct proof often reaches dead ends. I. Arwa Linjawi & I. Asma’a Ashenkity ...
... Direct proofs lead from the hypothesis of a theorem to the conclusion. They begin with the premises; continue with a sequence of deductions, and ends with the conclusion. Direct proof often reaches dead ends. I. Arwa Linjawi & I. Asma’a Ashenkity ...
Full text
... identified with a subgroup of F(G) and therefore F(G) has a free subgroup of infinite rank. T h e P r o o f of C o r o l l a r y 2: This follows from the Proposition. Indeed, assume that there exist only finitely many prime numbers, call them pi,P2? • • • ?Ps? such that whenever q is a prime number ...
... identified with a subgroup of F(G) and therefore F(G) has a free subgroup of infinite rank. T h e P r o o f of C o r o l l a r y 2: This follows from the Proposition. Indeed, assume that there exist only finitely many prime numbers, call them pi,P2? • • • ?Ps? such that whenever q is a prime number ...
12.3 Geometric Sequences Series
... Precalculus 12.3 Geometric Sequences; Geometric Series Objective: able to determine if a sequence is geometric; find a formula for a geometric sequence; find the sum of a geometric sequence (a geometric series). ...
... Precalculus 12.3 Geometric Sequences; Geometric Series Objective: able to determine if a sequence is geometric; find a formula for a geometric sequence; find the sum of a geometric sequence (a geometric series). ...
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.