Consecutive integers squares

The numbers 1, 2, 3, and 4 are four consecutive integers. The numbers 1, 4, 9, and 16 are four consecutive squares of those first four numbers. And the cubes of the same four numbers are 1, 8, 27, and 64. Squares and cubes of consecutive numbers are squared and cubed, respectively, before being added together or having some other operation performed upon them. 

First write the two consecutive integers as n and n + 1. The squares of those two numbers are n2 and (n + l)2. To write the equation, add the two squares together and set them equal to 145. Then do the squaring on the left, combine like terms, subtract 145 from each side to set the equation equal to 0, and solve the quadratic equation. 

Magic square is an unusual numerical configuration containing consecutive integers in arrangements so that the sum of numbers in any row, column, or diagonal are identical. Such squares were known approximately 4,000 years ago in China. 

The basic magic square is a square containing consecutive integers starting with number 1. Three of the basic magic squares are shown in Table 1. 

The sides of a rectangle are consecutive even integers. What is the longer side, if the area is 168 square centimeters ... 

Then in 1913 Moseley, working in Manchester, discovered what we now call atomic number (Moseley, 1913). He began by photographing the X-ray spectrum of 12 elements, 10 of which occupied consecutive places in the periodic table. He discovered that the frequencies of features called K-lines in the spectrum of each element were directly proportional to the square of the integer representing the position of each successive element in the table. As Moseley put it, here was proof that "there is in the atom a fundamental quantity, which increases by regular steps as we pass from one element to the next." This fundamental quantity, first referred to as "atomic number" in 1920 by Ernest Rutherford, is now identified as the munber of protons in the nucleus of an atom. Moseley s work provided a method that could be used to determine exactly how many empty spaces remained in the periodic table. 

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