k-TO-l FUNCTIONS ON ARCS FOR k EVEN 1. eitherf((x,p))çz(f(x),f(p))
... is finite, there is a positive number d' < d such that no point within d' of p maps to f(p) except p. Choose any number x' less than p so that |x' —p\ < d'. The set f'l(f(x')) is finite so there is an x with x' < x < p and f(x) = f(x') such that no point of (x, p) maps to f(x'). Part 1 is true for t ...
... is finite, there is a positive number d' < d such that no point within d' of p maps to f(p) except p. Choose any number x' less than p so that |x' —p\ < d'. The set f'l(f(x')) is finite so there is an x with x' < x < p and f(x) = f(x') such that no point of (x, p) maps to f(x'). Part 1 is true for t ...
Group action
... The proof is very similar to 4(b), so we shall say only about the differences. Firstly, the ring of Gaussian numbers is replaced by Z . Secondly, the modular arithmetic is slightly different: in the previous example, we were reducing a2 + b2 = 0 (mod p) to (a/b)2 = –1 (mod p); in a similar way ...
... The proof is very similar to 4(b), so we shall say only about the differences. Firstly, the ring of Gaussian numbers is replaced by Z . Secondly, the modular arithmetic is slightly different: in the previous example, we were reducing a2 + b2 = 0 (mod p) to (a/b)2 = –1 (mod p); in a similar way ...
Comparing Integers
... 36. Zero is greater than negative seven. 37. The opposite of negative nine is less than ten. 38. Writing A number is sometimes less than its opposite. Use examples to ...
... 36. Zero is greater than negative seven. 37. The opposite of negative nine is less than ten. 38. Writing A number is sometimes less than its opposite. Use examples to ...
ELEMENTARY NUMBER THEORY
... the Quadratic Reciprocity Law as a goal can be built up from Chapters 1 through 9. It is unlikely that every section in these chapters need be covered; some or all of Sections 5.4, 6.2, 6.3, 6.4, 7.4, 8.3, 8.4, and 9.4 can be omitted from the program without destroying the continuity in our developm ...
... the Quadratic Reciprocity Law as a goal can be built up from Chapters 1 through 9. It is unlikely that every section in these chapters need be covered; some or all of Sections 5.4, 6.2, 6.3, 6.4, 7.4, 8.3, 8.4, and 9.4 can be omitted from the program without destroying the continuity in our developm ...
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.