OPEN DIOPHANTINE PROBLEMS 1. Diophantine Equations 1.1
... Even the simplest case of quadratic forms suggests open problems. The determination of all positive integers which are represented by a given binary form is far from being solved. It is also expected that infinitely many real quadratic fields have class number one, but it is not even known that ther ...
... Even the simplest case of quadratic forms suggests open problems. The determination of all positive integers which are represented by a given binary form is far from being solved. It is also expected that infinitely many real quadratic fields have class number one, but it is not even known that ther ...
Geometric Sequence
... Sigma Notation – A series can be represented in a compact form, called summation notation, or sigma notation. The Greek capital letter sigma, , is used to indicate a sum. ...
... Sigma Notation – A series can be represented in a compact form, called summation notation, or sigma notation. The Greek capital letter sigma, , is used to indicate a sum. ...
(0.4) K -f, - American Mathematical Society
... are sequences of approximants for some continued fraction b0 + K^=xal,/b„1 The answer is that "almost any sequence is". The only conditions that have to be satisfied are/0 ¥= oo and/,_, ¥=f„ for all n > 1. For continued fractions b0 + K(an/l) the conditions are f0 ¥= ooand/,_, # f„,f„-\ ^f,,+ \ for ...
... are sequences of approximants for some continued fraction b0 + K^=xal,/b„1 The answer is that "almost any sequence is". The only conditions that have to be satisfied are/0 ¥= oo and/,_, ¥=f„ for all n > 1. For continued fractions b0 + K(an/l) the conditions are f0 ¥= ooand/,_, # f„,f„-\ ^f,,+ \ for ...
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.