Numeration 2016 - Katedra matematiky
... (i) For every α ∈ Cγ there exists some n ∈ N0 with α + n ∈ F0 (γ) ∪ F1 (γ). (ii) F1 (γ) consists of fundamental CNS bases for Z[γ]. Here the algebraic integer β is called a fundamental CNS basis for O if it satisfies the following properties: (1) β − n is a CNS basis for O for all n ∈ N0 . (2) β + 1 ...
... (i) For every α ∈ Cγ there exists some n ∈ N0 with α + n ∈ F0 (γ) ∪ F1 (γ). (ii) F1 (γ) consists of fundamental CNS bases for Z[γ]. Here the algebraic integer β is called a fundamental CNS basis for O if it satisfies the following properties: (1) β − n is a CNS basis for O for all n ∈ N0 . (2) β + 1 ...
Solutions - CMU Math
... the only way to get a number divisible by 3 by adding three of these is 1 + 1 + 1, so those scores must be entered first. Now we have an odd sum, so we must add 71 in order for the sum to be divisible by 4. That leaves 80 for the last score entered. 6. Find the number of integers n, 1 ≤ n ≤ 25 such ...
... the only way to get a number divisible by 3 by adding three of these is 1 + 1 + 1, so those scores must be entered first. Now we have an odd sum, so we must add 71 in order for the sum to be divisible by 4. That leaves 80 for the last score entered. 6. Find the number of integers n, 1 ≤ n ≤ 25 such ...
FERMAT`S LITTLE THEOREM 1. Introduction When we compute the
... list mod 7, so their products are the same modulo 7: 1 · 2 · 3 · 4 · 5 · 6 ≡ a · 2a · 3a · 4a · 5a · 6a ≡ a6 (1 · 2 · 3 · 4 · 5 · 6) mod 7. Canceling the common factors 1, 2, 3, 4, 5, and 6 from both sides, since they are all nonzero mod 7, we are left with 1 ≡ a6 mod 7. Remark 2.2. In the proof we ...
... list mod 7, so their products are the same modulo 7: 1 · 2 · 3 · 4 · 5 · 6 ≡ a · 2a · 3a · 4a · 5a · 6a ≡ a6 (1 · 2 · 3 · 4 · 5 · 6) mod 7. Canceling the common factors 1, 2, 3, 4, 5, and 6 from both sides, since they are all nonzero mod 7, we are left with 1 ≡ a6 mod 7. Remark 2.2. In the proof we ...
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.