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What is a perfect square?

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    * A square, sometimes called a perfect square, is the result of multiplying a number by itself as in N = nxn = n^2.

    * The perfect squares are the squares of the counting numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, etc. which produce 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, etc.

    * The last digit in the square of a number must be one of the following: 0, 1, 4, 5, 6, or 9. Note that there are numbers that end in these digits that are not squares but to be a square, they must end in one of these digits.

    * If the last digit of a number is 0, its square ends in 00 and the preceding digits form a square.

    * If the last digit of a number is 1 or 9, its square ends in 1 and the number formed by the preceding digits must be divisible by 4.

    * If the last digit of a number is 2 or 8, its square ends in 4 and the preceding digit must be even.

    * If the last digit of a nuber is 3 or 7, its square ends in 9 and the number formed by the preceding digits must be divisible by 4.

    * If the last digit of a number is 4 or 6, its square ends in 6 and the preceding digit must be odd.

    * If the last digit od a number is 5, its square ends in 25 and the preceding digits must be 0, 2, 06 or 56.

    * The last two digits of a 3 or more digit square number must be one of the following: 00, 01, 04, 09, 16, 21, 24,
    25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89, or 96.

    * The sum of the digits in a square MUST add up to 1, 4, 7, or 9.

    * The perfect squares are simply the successive sum of the odd numbers.
    0 + 1 = 1
    1 + 3 = 4
    1 + 3 + 5 = 9
    1 + 3 + 5 + 7 = 16
    1 + 3 + 5 + 7 + 9 = 25, etc.

    * The perfect squares can also be expressed by the sum of 1+1+2+2+3+3+....(n-1)+(n-1)+n. For example, 5^2 = 1+1+2+2+3+3+4+4+5 = 25.

    * The nth square can be determined from the previous two from n^2 = 2(n - 1)^2 - (n - 2)^2 + 2. For example, 6^2 = 25 + 25 - 16 + 2 = 36.

    * The nth square is equal to n^2.
    n....1....2....3....4....5....6....7....8....9....10
    Sq..1...4....9....16..25..36..49..64...81...100

    * The nth non-square is n + sqrt(n) r.o. (r.o. = rounded off)
    n....................1....2....3....4....5....6....7....8....9....10....11....12....13....14....15
    Non-square.....2....3....4....6....7....8...10..11...12...13....14....15....17....18....19
    6th non-square = 6 + 2[.449] = 8; 12th non-square = 12 + 3[.464] = 15.

    * The perfect squares are also the sum of consecutive triangular numbers, Tn = n(n + 1)/2 = 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, etc.
    1^2 = 0 + 1 = 1
    2^2 = 1 + 3 = 1 + 3
    3^2 = 1 + 3 + 5 = 3 + 6
    4^2 = 1 + 3 + 5 + 7 = 6 + 10
    5^2 = 1 + 3 + 5 + 7 + 9 = 10 + 15 - 1, 3, 6, 10, 15, etc., being the sequential triangular numbers.

    * All even squares are congruent to 0 modulo 4 meanig that all even squares minus 0 are divisible by 4. More simplistically, all even squares are evenly divisible by 4.

    * All odd squares are congruent to 1 modulo 8 meaning that all odd squares minus 1 are evenly divisible by 8. Another way of staing this is that the square of an odd number is always of the form 8n + 1.

    * The square of a number is either divisible by 4 or leaves a remainder of 1 when divided by 4.

    * The square of an odd number is always of the form 8n + 1.

    * The sum of the first n squares, i.e., 1^2 + 2^2 + 3^2 + 4^2 + .....+ n^2 = n(n + 1)(2n + 1)/6.

    * An easy way to determine the square of N, s to find two numbers whose mean is N from which N^2 = (N+1)(N-1) + 1. For example, for N = 25, 25 is the mean of 24 and 26. Therefore, the square of 25 is 24x26 + 1 = 625.

    * An easy way to derive the squares: Having x^2, the value of (x + 1)^2 can be expressed by (x + 1)^2 = x^2 +
    (2x + 1). What does this mean? If you know the square of a particular number, say 30^2 = 900, then the square
    of 31 is 31^2 = 30^2 + (2(30) + 1) = 900 + 60 + 1 = 961. Similarly, the square of (x - 1) is defined by (x - 1)^2 =
    x^2 -2x + 1 and for 29^2 we have 29^2 = 900 - 60 + 1 = 841. We also know that (x + 2)^2 = x^2 + 4x + 4 and (x
    + 3)^2 = x^2 + 6x + 9. This can be easily written as (x + n)^2 = x^2 + (2xn + n^2) and in the case of (x - n)^2 =
    x^2 -(2xn + n^2). Therefore, knowing the simple squares such as 20^2, 30^2, 70^2 90^2, 120^2, etc., it is an
    easy computation to arrive at the adjacent 5 squares.

    *--Did you ever notice anything odd about the series of squares? 1-4-9-16-25-36-49-64-81-100-121-144-169-etc.
    Squares - 1 - 4 - 9 - 16 - 25 - 36 - 49 - 64 - 81 - 100 - 121 - 144 - 169
    Differences - 3 5 7 9 11 13 15 17 19 21 23 25 and so on ad infinatum. If you happen to
    know two adjacent squares, such as 15^2 = 225 and 16^2 = 256, you immediately know the next squares on
    either side of the two known ones. If you wanted to know the square of 23, you could use the expression (x +
    n)^2 = x^2 + (2n + n^2) = (20 + 3)^2 = 20^2 + (2(20)3 + 9) = 400 + 129 = 529 or the same expression to find (x
    + 1)^2 = 441, obtain the difference from 441 and 400, and apply it to (x + 3)^2 = 400 + 41 + 43 + 45 = 529.

    * Having one square, x, we know that the next square is (x + 1)^2 = x^2 + (2x + 1).

    * The square of any number ending in 5 is obtained from x(x + 1) and placing a 25 after the result, e.g., 25^2 =
    2(2 + 1) plus 25 = 6 + 25 = 625.

    * If an integer N>1 is not a perfect square, then sqrtN is irrational, i.e., sqrtN cannot be expressed as a/b, where a and b are integers.

    * Any integer which is a ratio of squares is, itself, a square.

    * If both a and b are perfect squares, then a(b) is a perfect square.

    * There exists sets of 4 integers such that the sum of all four of them and the sum of each pair of them are squares.
    Verify with 386, 2114, 3970 and 10,430.

    * Lagrange's Theorem - Any whole number can be decomposed into sums of up to four squares.
    Example - 97 = 81 + 16 = 64 + 25 + 4 + 4.

    * 1/10th of the first 100 integers are perfect squares. 1/100th of the first 10,000 numbers are perfect squares. 1/1000th of the first 1,000,000 numbers are perfect squares.

    * The product of three consecutive integers is never a square.

    * An easy way of determining the square of a number is to identify two numbers that are the mean of the number in question, multiply them together and add the square of the distance from the mean.
    Example: 24^2 = (23x25) + 1^2 = 576 or 16^2 = (14x18) + 2^2 = 196.

    * Perhaps obvious:

    The squares of odd square numbers are odd and the squares of even square numbers are even.
    Conversely, the square roots of odd numbers are odd and the square roots of even nubers are even.

    * While not very useful in calculating squares, it is worth noting that a square is the sum of 1+1+2+2+3+3+...(n-1) + (n-1)+n.
    Example: 5^2 = 1+1+2+2+3+3+4+4+5 = 25.

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