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In mathematics, a square number or perfect square is an integer that is the square of an integer; [1] in other words, ... For example, 5 2 = 25 = 1 + 3 + 5 + 7 + 9.
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 ...
Bernard Frénicle de Bessy (c. 1604 – 1674), was a French mathematician born in Paris, who wrote numerous mathematical papers, mainly in number theory and combinatorics.He is best remembered for Des quarrez ou tables magiques, a treatise on magic squares published posthumously in 1693, in which he described all 880 essentially different normal magic squares of order 4.
Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares? Are there infinitely many primes p such that p − 1 is a perfect square? In other words: Are there infinitely many primes of the form n 2 + 1? As of 2025, all four problems are unresolved.
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.
Lastly the four rhomboids that form elongated crosses also give the magic sum: 23+1+9+24+8, 15+1+17+20+12, 14+1+18+13+19, 7+1+25+22+10. Such squares with 1 at the center cell are also called God's magic squares in Islamic amulet design, where the center cell is either left blank or filled with God's name. [26]
Not only so, but the proportionate number of squares diminishes as we pass to larger numbers, Thus up to 100 we have 10 squares, that is, the squares constitute 1/10 part of all the numbers; up to 10000, we find only 1/100 part to be squares; and up to a million only 1/1000 part; on the other hand in an infinite number, if one could conceive of ...
So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. [ 2 ] [ 4 ] Euclid proved c. 300 BCE that every prime expressed as M p = 2 p − 1 has a corresponding perfect number M p × ( M p +1)/2 = 2 p − 1 × (2 p − 1) .