Square number

A square number is a number that results from the multiplication of an integer by itself. For example, a square. The first square numbers are

For some authors, the zero is not a perfect square, so that the sequence of numbers begins with the first one.

The name square number is derived from the geometric figure of the square. The number of stones that you need for laying a square is always a square number. Thus, for example, place a square of side length 4 with the help of 16 stones.

Because of this relationship with a geometric figure count the squares to the figurate numbers, which include the triangular numbers and cubic numbers belong. These terms were already known to Greek mathematicians of antiquity.

Properties

Just square numbers are the square of even numbers, while odd square numbers are the square of odd numbers.

Formulas to generate square numbers

Each number is the square sum of the first odd natural numbers.

This law, known in English-language literature as Odd Number Theorem, is illustrated by the following pictures.

From left to right here are the first four square numbers are represented by the corresponding number of balls. The blue balls indicate, respectively, the difference to the previous square number. As from left to right always added one row and one row, the number of blue balls increased by 2 Starting with the one on the far left through the blue balls so all odd numbers.

The Education Act can also be proved directly using the first binomial formula. For this, the corresponding sums are represented by the formula

Shown. By induction it can be shown to be valid. The induction starts

For is obviously correct. Assuming that

Applies, is then the induction conclusion

Valid.

Each square number is twice the sum of the first natural numbers plus the number.

Trick to calculate five - square numbers in your head

The square of numbers ending in 5, can be easily calculated in the head. You multiply the number without the digit 5 (eg 65, 6 ) with its successor ( here 6 1 = 7) and depends on the product (in this case 6 x 7 = 42) paragraphs 2 and 5 of (final result 4225 ).

Proof: A five-digit can be represented as. Your square is thus.

Relations to other figurate numbers

Triangular numbers

Each square number can be represented as the sum of two consecutive triangular numbers. The picture shows an example of how the square number 25 is the sum of triangular numbers and.

This phenomenon can also be described by a formula.

Centered square numbers

In addition to the square numbers underlying patterns, there is a second pattern to create a square. This will put more squares around a stone in the middle of the square. The need for this pattern number of stones corresponds to a centered square number. Each centered square number is the sum of two consecutive square numbers, as can be seen in the geometric pattern.

The formula for centered square numbers can be switched so that the two squares are visible with the aid of the first binomial formula.

Pyramid numbers

The sum of the first square numbers results in the -th pyramidal number.

The following figure illustrates this relationship using the example of the fourth pyramidal number.

Final digits of square numbers

Square numbers never end with one of the digits 2, 3, 7 or 8, since no square of a single digit number ends with one of these numbers.

If the last digit of any number, then for whose square

The last digit of is thus identical to the last digit of. One of the first square numbers 0, 1, 4, 9, 16, 25, 36, 49, 64 and 81, however, is not a number which ends 2, 3, 7 or 8.

Divider number

Only square numbers have an odd number of divisors. Proof: Let, and. It is because. contains all divisors of, so is the number of divisors of the same. Is a square number, then. Otherwise.

Series of the reciprocals

→ Main article: Basler problem

Is the sum of the reciprocal values ​​of all the squares

It was a long time not known whether this series converges, and if so, to what limit. Only Leonhard Euler found the value of the series in 1735.

Sums of consecutive square numbers

There are some strange relations for the sum of consecutive square numbers:

Or generally

Some primes can be described as the sum of two, three or even six consecutive squares write ( other numbers of summands are not possible):