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Square number. Square number 16 as sum of gnomons. In mathematics, a square number or perfect square is an integer that is the square of an integer; [1] in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals 32 and can be written as 3 × 3.
A perfect square is an element of algebraic structure that is equal to the square of another element. Square number, a perfect square integer. Entertainment
The difference of two squares is used to find the linear factors of the sum of two squares, using complex number coefficients. For example, the complex roots of can be found using difference of two squares: (since ) Therefore, the linear factors are and . Since the two factors found by this method are complex conjugates, we can use this in ...
Perfect number. In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, since 1 + 2 + 4 + 7 + 14 = 28.
The following is a list of all currently known Mersenne primes and perfect numbers, along with their corresponding exponents p. As of 2023, there are 51 known Mersenne primes (and therefore perfect numbers), the largest 17 of which have been discovered by the distributed computing project Great Internet Mersenne Prime Search, or GIMPS. [2]
When a triple of numbers a, b and c forms a primitive Pythagorean triple, then (c minus the even leg) and one-half of (c minus the odd leg) are both perfect squares; however this is not a sufficient condition, as the numbers {1, 8, 9} pass the perfect squares test but are not a Pythagorean triple since 1 2 + 8 2 ≠ 9 2. At most one of a, b, c ...
For non-generalized pentagonal numbers, in addition to the perfect square test, it is also required to check if 24 x + 1 ≡ 5 mod 6 {\displaystyle {\sqrt {24x+1}}\equiv 5\mod 6} The mathematical properties of pentagonal numbers ensure that these tests are sufficient for proving or disproving the pentagonality of a number.
Contents. (Top) Superperfect number. In number theory, a superperfect number is a positive integer n that satisfies. where σ is the divisor summatory function. Superperfect numbers are not a generalization of perfect numbers but have a common generalization. The term was coined by D. Suryanarayana (1969).