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A non-negative integer is a square number when its square root is again an integer. For example, =, so 9 is a square number. A positive integer that has no square divisors except 1 is called square-free. For a non-negative integer n, the n th square number is n 2, with 0 2 = 0 being the zeroth one. The concept of square can be extended to some ...
All centered square numbers and their divisors have a remainder of 1 when divided by 4. Hence all centered square numbers and their divisors end with digit 1 or 5 in base 6, 8, and 12. Every centered square number except 1 is the hypotenuse of a Pythagorean triple (3-4-5, 5-12-13, 7-24-25, ...). This is exactly the sequence of Pythagorean ...
A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.
Proof that centered octagonal numbers are odd squares. Each centered k-gonal number in the series is k times the previous triangular number, plus 1. This can be formalized by the expression (+) +, where n is the series rank, starting with 0 for the initial 1. For example, each centered square number in the series is four times the previous ...
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A square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes. From Gulley (2010). The n th coloured region shows n squares of dimension n by n (the rectangle is 1 evenly divided square), hence the area of the n th region is n times n × n.
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The numbers are based on a $50 a square game, with a $625 payout for the 1st and 3rd quarters, a $1,250 payout for halftime, and a $2,500 payout for the end of the game. (The cells are colored ...
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