Generalization of the Genocchi Numbers to their q-analogue Matthew Rogala April 15, 2008
... The Gamma function was introduced in a series of letters between Leonhard Euler and Christian Goldbach in the year 1729. It came about as a solution to a problem many mathematicians were pondering at the time: a way to interpolate the factorials. What is meant by ”interpolation” is best shown via ex ...
... The Gamma function was introduced in a series of letters between Leonhard Euler and Christian Goldbach in the year 1729. It came about as a solution to a problem many mathematicians were pondering at the time: a way to interpolate the factorials. What is meant by ”interpolation” is best shown via ex ...
Grade 6 Math Circles Sequences What is a Sequence?
... its elements any more. We say that an item in a sequence is a term. We will often ask you to find a certain term of a sequence, for example, the 3rd term, or the 20th term. We write t3 for the 3rd term, and t20 for the 20th term. In general, tn is the nth term, where n is the ...
... its elements any more. We say that an item in a sequence is a term. We will often ask you to find a certain term of a sequence, for example, the 3rd term, or the 20th term. We write t3 for the 3rd term, and t20 for the 20th term. In general, tn is the nth term, where n is the ...
Full text
... In one of his famous results, Fermat showed that there exists no Pythagorean triangle with integer sides whose area is an integer square. His elegant method of proof is one of the first known examples in the history of the theory of numbers where the method of infinite descent is employed. Mohanty [ ...
... In one of his famous results, Fermat showed that there exists no Pythagorean triangle with integer sides whose area is an integer square. His elegant method of proof is one of the first known examples in the history of the theory of numbers where the method of infinite descent is employed. Mohanty [ ...
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.