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In mathematics, a square number or perfect square is an integer that is the square of an integer; [1] ... 11 2 = 121 12 2 = 144 13 2 = 169 14 2 = 196 15 2 = 225 16 2 ...
The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial x + 1 is the quadratic polynomial (x + 1) 2 = x 2 ...
A perfect square is an element of algebraic structure that is equal to the square of another element. ... This page was last edited on 21 February 2024, at 11:30 (UTC).
For the perfect numbers, define P n = 2 n−1 M n be the value which is perfect if M n is prime. When n = 4 k +1 and k > 0 , 2 4 k ≡ 16 (mod 20) , so P n ≡ 16×11 ≡ 16 (mod 20) and will end in 16, 36, 56, 76 or 96.
Every even perfect number ends in 6 or 28, base ten; and, with the only exception of 6, ends in 1 in base 9. [55] [56] Therefore, in particular the digital root of every even perfect number other than 6 is 1. The only square-free perfect number is 6. [57]
Brocard's problem is a problem in mathematics that seeks integer values of such that ! + is a perfect square, where ! is the factorial. Only three values of n {\displaystyle n} are known — 4, 5, 7 — and it is not known whether there are any more.
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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} .