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... conditions demand that the modulus of the leading weight coefficient a0 should be sufficiently large (see also §5). There are many sequences that satisfy these conditions. For example, take an arbitrary holomorphic function hx(z) defined in a neighborhood of zero. Then the sequence appearing as the ...
... conditions demand that the modulus of the leading weight coefficient a0 should be sufficiently large (see also §5). There are many sequences that satisfy these conditions. For example, take an arbitrary holomorphic function hx(z) defined in a neighborhood of zero. Then the sequence appearing as the ...
A Journey into Triangular Number Land
... club member for help if you’re not sure how to do this.) Then, see if you can find the diagonals within Pascal’s triangle which show the triangular numbers. What interesting patterns are shown in the other diagonals of Pascal’s triangle? Some interesting properties of triangular numbers: 1. Lets den ...
... club member for help if you’re not sure how to do this.) Then, see if you can find the diagonals within Pascal’s triangle which show the triangular numbers. What interesting patterns are shown in the other diagonals of Pascal’s triangle? Some interesting properties of triangular numbers: 1. Lets den ...
Elementary Number theory and Cryptography Epiphany 2011
... — Here the product of α1 α2 ᾱ3 and its complex conjugate again gives 1105; then α1 α2 ᾱ3 = 23 + 24i and 232 + 242 = 1105. — Here the product of α1 ᾱ2 ᾱ3 and its complex conjugate gives 1105; then α1 ᾱ2 ᾱ3 = 33 + 4i and 332 + 42 = 1105. 7. Prove that the natural number n has as many representat ...
... — Here the product of α1 α2 ᾱ3 and its complex conjugate again gives 1105; then α1 α2 ᾱ3 = 23 + 24i and 232 + 242 = 1105. — Here the product of α1 ᾱ2 ᾱ3 and its complex conjugate gives 1105; then α1 ᾱ2 ᾱ3 = 33 + 4i and 332 + 42 = 1105. 7. Prove that the natural number n has as many representat ...
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.