PROBLEM SET 7
... (It can also be formulated in terms of pigeons and pigeonholes, hence the name.) The proof of this pigeonhole principle is easy. It is more difficult to know when to apply it. There are many surprising applications of the pigeonhole principle. The pigeonhole principle was first explicitly formulated ...
... (It can also be formulated in terms of pigeons and pigeonholes, hence the name.) The proof of this pigeonhole principle is easy. It is more difficult to know when to apply it. There are many surprising applications of the pigeonhole principle. The pigeonhole principle was first explicitly formulated ...
Discrete Mathematics I Lectures Chapter 4
... particular but arbitrarily chosen m for which P(m) is true.” Write down a clear statement for what you need to prove: “Therefore, Q(m) is true” On the side, write down every definition and axiom that you know which you think might be relevant From the starting point, use the definitions and axioms t ...
... particular but arbitrarily chosen m for which P(m) is true.” Write down a clear statement for what you need to prove: “Therefore, Q(m) is true” On the side, write down every definition and axiom that you know which you think might be relevant From the starting point, use the definitions and axioms t ...
MA/CSSE 473 Day 9 Announcements and Summary
... a. Fermat’s Little Theorem: If p is prime, and a is not 0 (mod p), then ap-1 ≡ 1 (mod p). i. So if we find an a in range 1 < a < N for which aN-1 ≢ 1 (mod N), the number is not prime. ii. But it is possible that N is composite but there is an a with aN-1 ≡ 1 (mod N). iii. Such an a is called a Ferma ...
... a. Fermat’s Little Theorem: If p is prime, and a is not 0 (mod p), then ap-1 ≡ 1 (mod p). i. So if we find an a in range 1 < a < N for which aN-1 ≢ 1 (mod N), the number is not prime. ii. But it is possible that N is composite but there is an a with aN-1 ≡ 1 (mod N). iii. Such an a is called a Ferma ...
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.