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Legendre's three-square theorem states which numbers can be expressed as the sum of three squares; Jacobi's four-square theorem gives the number of ways that a number can be represented as the sum of four squares. For the number of representations of a positive integer as a sum of squares of k integers, see Sum of squares function.
A most-perfect magic square of order n is a magic square containing the numbers 1 to n 2 with two additional properties: Each 2 × 2 subsquare sums to 2s, where s = n 2 + 1. All pairs of integers distant n/2 along a (major) diagonal sum to s.
The number of ways to write a natural number as sum of two squares is given by r 2 (n). It is given explicitly by = (() ()) where d 1 (n) is the number of divisors of n which are congruent to 1 modulo 4 and d 3 (n) is the number of divisors of n which are congruent to 3 modulo 4. Using sums, the expression can be written as:
This sum can also be found in the four outer numbers clockwise from the corners (3+8+14+9) and likewise the four counter-clockwise (the locations of four queens in the two solutions of the 4 queens puzzle [50]), the two sets of four symmetrical numbers (2+8+9+15 and 3+5+12+14), the sum of the middle two entries of the two outer columns and rows ...
square number is 1 (solve the Diophantine equation x 2 = y 3 + 4y, where y is even); generalized pentagonal number is 171535 (solve the Diophantine equation x 2 = y 3 + 144y + 144, where y is divisible by 12); tetrahedral number is 2925. Note that 0 and 1 are the only normal magic constants of rational order which are also rational squares.
The sum of all numbers on each circle (not including 9) = 2 × 69; There exist 8 semicircles, where the sum of numbers = magic number 69; there are 16 line segments (semicircles and radii) with magic number 69, more than a 6 order magic square with only 12 magic numbers.
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 3 2 and can be written as 3 × 3.
An associative magic square is a magic square for which each pair of numbers symmetrically opposite to the center sum up to the same value. For an n × n square, filled with the numbers from 1 to n 2 , this common sum must equal n 2 + 1.