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A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...
The square root of a positive integer is the product of the roots of its prime factors, because the square root of a product is the product of the square roots of the factors. Since p 2 k = p k , {\textstyle {\sqrt {p^{2k}}}=p^{k},} only roots of those primes having an odd power in the factorization are necessary.
The two square roots of a negative number are both imaginary numbers, and the square root symbol refers to the principal square root, the one with a positive imaginary part. For the definition of the principal square root of other complex numbers, see Square root § Principal square root of a complex number.
A simple case arises in taking the square root of a positive real number. For example, 4 has two square roots: 2 and −2; of these the positive root, 2, is considered the principal root and is denoted as 4 . {\displaystyle {\sqrt {4}}.}
Aside from being the smallest square triangular number other than 1, it is also the only triangular number (other than 1) whose square root is also a triangular number. 36 is also the eighth refactorable number, as it has exactly nine positive divisors, and 9 is one of them; [3] in fact, it is the smallest positive integer with at least nine ...
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The main difficulty is that, in order to solve the problem, the square-roots should be computed to a high accuracy, which may require a large number of bits. The problem is mentioned in the Open Problems Garden. [4] Blomer [5] presents a polynomial-time Monte Carlo algorithm for deciding whether a
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