Unsolved problems in mathematics | Integer sequences | Recurrence relations
In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29. The numerators of the same sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82. Both the Pell numbers and the companion Pell numbers may be calculated by means of a recurrence relation similar to that for the Fibonacci numbers, and both sequences of numbers grow exponentially, proportionally to powers of the silver ratio 1 + √2. As well as being used to approximate the square root of two, Pell numbers can be used to find square triangular numbers, to construct integer approximations to the right isosceles triangle, and to solve certain combinatorial enumeration problems. As with Pell's equation, the name of the Pell numbers stems from Leonhard Euler's mistaken attribution of the equation and the numbers derived from it to John Pell. The Pell–Lucas numbers are also named after Édouard Lucas, who studied sequences defined by recurrences of this type; the Pell and companion Pell numbers are Lucas sequences. (Wikipedia).
2023 Number Challenge: Pell's Equation with d=2023
Check out other 2023 Number Challenges from this list. Share with your friends!! https://www.youtube.com/playlist?list=PLXpXgWDr4HM7KKeX7CaQIu4tfPRJ2HiUM Please subscribe to the channel. In this video, we study Pell's equation when d=2023. We use two methods: 1. With continue continu
From playlist Math Problems with Number 2023
Triangular numbers, Venn diagrams and probability.
From playlist PIXL PPE Paper 2 June 2016 Higher Tier AQA Style Worked Solutions
[ANT08] Continued fractions, Pell's equation, and units of Z[√d]
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From playlist [ANT] An unorthodox introduction to algebraic number theory
👉 Learn all about decimals. Decimals are numbers written with a decimal point. Digits can be written to the right or to the left of the decimal point. Digits are written to the left of the decimal point increase in value by multiples of 10 while digits written to the right decrease by mul
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#shorts This video shows a pattern when squaring number with digits of 1.
From playlist Math Shorts
#MegaFavNumbers, Mine is 226,153,980. What's yours?
#MegaFavNumbers Welcome to MainlyMathExcursions! Today we'll be taking a little trip Pellward. We'll board the bus at 61 and arrive at 226,153,980 and 1,766,319,049. This is my first YouTube video, I hope you like it. Here's a shout-out to blackpenredpen for his encouragement, and to ever
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Interesting Math Facts: Forty, One, and Palindrome Numbers
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This video gives a precise definition of a decimal number as a special kind of rational number; one for which there is an expression a/b where a and b are integers, with b a power of ten. For such a number we can extend the Hindu-Arabic notation for integers by introducing the decimal form
From playlist Math Foundations
#MegaFavNumbers A small discussion regarding a famous "proof" that all numbers are interesting. This video is my participation in the #MegaFavNumbers project.
From playlist MegaFavNumbers
Algebra - Ch. 6: Factoring (6 of 55) How to Determine a Prime Number? NOTE: 49 IS NOT A PRIME NUMBER
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From playlist THE "HOW TO" PLAYLIST
Pell Number Lattice Path Enumeration and visual recurrence formula (synthwave; visual proof)
This synthwave enumeration shows all of the lattice paths with steps (1,1), (-1,1) and (2,0) from the origin (0,0) to the line y = n-1. The number of such lattice paths is counted by the Pell numbers, and we use the visual enumeration to show how to produce the (linear) recurrence formula
From playlist Discrete Mathematics Course
SM64 Pendulum Math [EP1]: 6,248,970 and 507,780 (#MegaFavNumbers)
Pannenkoek2012's Pendulum Video: https://www.youtube.com/watch?v=f35hBQ0zfO0 Pell's equation: https://en.wikipedia.org/wiki/Pell%27s_equation Brahmagupta's identity: https://en.wikipedia.org/wiki/Brahmagupta%27s_identity Pendulum Amplitude Equation: https://www.desmos.com/calculator/qimiwh
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[ANT08b] Square triangular numbers
Another application of Pell's equation.
From playlist [ANT] An unorthodox introduction to algebraic number theory
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