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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 ...
A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.
Below are listed the first prime numbers of many named forms and types. ... Primes p for which p − 1 divides the square of the product of all earlier terms. 2, 3, 5 ...
Babylonian tablet (c. 1800–1600 BCE), showing an approximation of √ 2 (1 24 51 10 in sexagesimal) in the context of the Pythagorean theorem for an isosceles triangle. Written mathematics began with numbers expressed as tally marks, with each tally representing a single unit. Numerical symbols consisted probably of strokes or notches cut in ...
A square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes. This shows that the square of the n th triangular number is equal to the sum of the first n cube numbers. Also, the square of the n th triangular number is the same as the sum of the cubes of the integers 1 to n.
[7] [8] [9] It is also not known if any odd perfect numbers exist; various conditions on possible odd perfect numbers have been proven, including a lower bound of 10 1500. [10] The following is a list of all currently known Mersenne primes and perfect numbers, along with their corresponding exponents p.
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Consequently, a square number is also triangular if and only if + is square, that is, there are numbers and such that =. This is an instance of the Pell equation x 2 − n y 2 = 1 {\displaystyle x^{2}-ny^{2}=1} with n = 8 {\displaystyle n=8} .