A Piece of the Pi: mathematics explained

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Number Theory 🔢

Repunits and prime numbers
A repunit (“repeated unit”) is a number that only contains the digit 1 in some number base.
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Richard Green
5
Ulam words and the Ulam sequence
The Ulam sequence is a sequence of positive integers xn, where x1=1, x2=2, and where each xn for n > 2 is defined to be the smallest integer that can be…
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Richard Green
5
The comma sequence
The comma sequence (1 , 12 , 35 , 94 , 135 , 186 , 248 , …) is so called because the difference between successive terms is given by the digits on…
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Richard Green
5
Egyptian fractions
The ancient Egyptians had notation to represent the fraction 1/n, but not for every fraction of the form a/b. Because of this, they typically…
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Richard Green
6
Magic squares of powers
A magic square of order n is an n by n grid of integers in which each row, each column, and each of the two main diagonals adds up to the same number.
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Richard Green
Generalizations of Hofstadter’s Q-sequence
The Fibonacci numbers are the famous sequence that begins 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.
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Richard Green
7
Partitions and primes
The picture above shows all the partitions of the numbers 1 up to 8.
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Richard Green
1
Continued fractions
A continued fraction is an expression obtained by representing a number as the sum of its integer part and the reciprocal of another number, and then…
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Richard Green
5
Arranging queens on a chessboard
What is the minimum number of chess pieces of a given type needed to attack every vacant square of the board? The answer turns out to be 9 for kings, 5…
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Richard Green
3
Friezes and Catalan numbers
A Conway–Coxeter frieze is an infinitely wide checkerboard of positive integers, such as the one in the picture above.
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Richard Green
The arithmetic derivative
Can you differentiate a number and get a nonzero answer?
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Richard Green
Friedman numbers
A Friedman number is a positive integer that can be written (nontrivially) using its own digits, together with parentheses and the operations of…
 • 
Richard Green
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